Nuprl Lemma : and_functionality_wrt_uiff3
∀[P1,P2,Q1,Q2:ℙ]. ((P1 = P2 ∈ ℙ) ⇒ {uiff(Q1;Q2)} ⇒ {P1 ∧ Q1 ⇐⇒ P2 ∧ Q2})
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
uiff_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
independent_isectElimination,
hypothesis,
cut,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
hypothesisEquality,
productEquality,
cumulativity,
introduction,
extract_by_obid,
isectElimination,
instantiate,
universeEquality
Latex:
\mforall{}[P1,P2,Q1,Q2:\mBbbP{}]. ((P1 = P2) {}\mRightarrow{} \{uiff(Q1;Q2)\} {}\mRightarrow{} \{P1 \mwedge{} Q1 \mLeftarrow{}{}\mRightarrow{} P2 \mwedge{} Q2\})
Date html generated:
2016_10_21-AM-09_35_02
Last ObjectModification:
2016_07_12-AM-04_59_40
Theory : core_2
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