Nuprl Lemma : ps-sigma-elim-equality-rule2
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}].
∀[t1:{X.A.B ⊢ _:(T)SigmaUnElim}]. ∀[t2:{X.Σ A B ⊢ _:T}].
  (t1)SigmaElim = t2 ∈ {X.Σ A B ⊢ _:T} supposing t1 = (t2)SigmaUnElim ∈ {X.A.B ⊢ _:(T)SigmaUnElim}
Proof
Definitions occuring in Statement : 
sigma-unelim-pscm: SigmaUnElim
, 
sigma-elim-pscm: SigmaElim
, 
presheaf-sigma: Σ A B
, 
psc-adjoin: X.A
, 
pscm-ap-term: (t)s
, 
presheaf-term: {X ⊢ _:A}
, 
pscm-ap-type: (AF)s
, 
presheaf-type: {X ⊢ _}
, 
ps_context: __⊢
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
equal: s = t ∈ T
, 
small-category: SmallCategory
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
squash: ↓T
, 
true: True
, 
presheaf-term-at: u(a)
Lemmas referenced : 
pscm-ap-term_wf, 
psc-adjoin_wf, 
ps_context_cumulativity2, 
presheaf-type-cumulativity2, 
presheaf-sigma_wf, 
sigma-unelim-pscm_wf, 
presheaf-term_wf2, 
pscm-ap-type_wf, 
presheaf-type_wf, 
small-category-cumulativity-2, 
ps_context_wf, 
small-category_wf, 
ps-sigma-elim-rule, 
equal_wf, 
squash_wf, 
true_wf, 
istype-universe, 
I_set_wf, 
presheaf-term-equal, 
ps-sigma-unelim-elim-term, 
presheaf-term-at_wf, 
cat-ob_wf, 
presheaf-term_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
equalityIstype, 
because_Cache, 
hypothesisEquality, 
cut, 
thin, 
instantiate, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
applyEquality, 
hypothesis, 
sqequalRule, 
universeIsType, 
applyLambdaEquality, 
hyp_replacement, 
equalitySymmetry, 
lambdaEquality_alt, 
imageElimination, 
equalityTransitivity, 
universeEquality, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
functionExtensionality, 
independent_isectElimination
Latex:
\mforall{}[C:SmallCategory].  \mforall{}[X:ps\_context\{j:l\}(C)].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[T:\{X.\mSigma{}  A  B  \mvdash{}  \_\}].
\mforall{}[t1:\{X.A.B  \mvdash{}  \_:(T)SigmaUnElim\}].  \mforall{}[t2:\{X.\mSigma{}  A  B  \mvdash{}  \_:T\}].
    (t1)SigmaElim  =  t2  supposing  t1  =  (t2)SigmaUnElim
Date html generated:
2020_05_20-PM-01_33_09
Last ObjectModification:
2020_04_03-AM-11_03_16
Theory : presheaf!models!of!type!theory
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