Nuprl Lemma : and-iff
∀[A:ℙ]. ∀[B,C:⋂a:A. ℙ]. ((A ⇒ (B ⇐⇒ C)) ⇒ {A ∧ B ⇐⇒ A ∧ C})
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
independent_functionElimination,
productEquality,
cumulativity,
hypothesisEquality,
cut,
rename,
isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionEquality,
lemma_by_obid,
applyEquality,
lambdaEquality,
isectEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbP{}]. \mforall{}[B,C:\mcap{}a:A. \mBbbP{}]. ((A {}\mRightarrow{} (B \mLeftarrow{}{}\mRightarrow{} C)) {}\mRightarrow{} \{A \mwedge{} B \mLeftarrow{}{}\mRightarrow{} A \mwedge{} C\})
Date html generated:
2016_05_13-PM-03_13_08
Last ObjectModification:
2016_01_06-PM-05_23_28
Theory : core_2
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