Nuprl Lemma : subtype_rel_functionality_wrt_iff
∀[A,B,C,D:Type]. ({uiff(A ⊆r B;C ⊆r D)}) supposing (B ≡ D and A ≡ C)
Proof
Definitions occuring in Statement :
ext-eq: A ≡ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
universe: Type
Definitions unfolded in proof :
guard: {T},
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
subtype_rel_transitivity,
subtype_rel_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
axiomEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeEquality
Latex:
\mforall{}[A,B,C,D:Type]. (\{uiff(A \msubseteq{}r B;C \msubseteq{}r D)\}) supposing (B \mequiv{} D and A \mequiv{} C)
Date html generated:
2016_05_13-PM-03_19_08
Last ObjectModification:
2015_12_26-AM-09_07_52
Theory : subtype_0
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