Nuprl Lemma : two-implies-quotient-true
∀[P,Q,R:ℙ]. ((P
⇒ Q
⇒ R)
⇒ {⇃(P)
⇒ ⇃(Q)
⇒ ⇃(R)})
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y]
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
guard: {T}
,
implies: P
⇒ Q
,
true: True
Definitions unfolded in proof :
guard: {T}
,
uall: ∀[x:A]. B[x]
,
implies: P
⇒ Q
,
member: t ∈ T
,
prop: ℙ
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
uimplies: b supposing a
,
quotient: x,y:A//B[x; y]
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
subtype_rel: A ⊆r B
,
true: True
Lemmas referenced :
quotient_wf,
true_wf,
equiv_rel_true,
quotient-member-eq,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
pointwiseFunctionalityForEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
hypothesis,
because_Cache,
independent_isectElimination,
pertypeElimination,
productElimination,
dependent_functionElimination,
applyEquality,
functionEquality,
independent_functionElimination,
natural_numberEquality,
productEquality,
universeEquality
Latex:
\mforall{}[P,Q,R:\mBbbP{}]. ((P {}\mRightarrow{} Q {}\mRightarrow{} R) {}\mRightarrow{} \{\00D9(P) {}\mRightarrow{} \00D9(Q) {}\mRightarrow{} \00D9(R)\})
Date html generated:
2016_05_14-AM-06_08_51
Last ObjectModification:
2015_12_26-AM-11_48_23
Theory : quot_1
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