Nuprl Lemma : predicate_rev_implies_weakening
∀[T:Type]. ∀[P1,P2:T ⟶ ℙ]. (P1 ⇐⇒ P2 ⇒ P1 ⇐ P2)
Proof
Definitions occuring in Statement :
predicate_equivalent: P1 ⇐⇒ P2,
predicate_rev_implies: P1 ⇐ P2,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
predicate_rev_implies: P1 ⇐ P2,
predicate_equivalent: P1 ⇐⇒ P2,
predicate_implies: P1 ⇒ P2,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
applyEquality,
hypothesisEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P1,P2:T {}\mrightarrow{} \mBbbP{}]. (P1 \mLeftarrow{}{}\mRightarrow{} P2 {}\mRightarrow{} P1 \mLeftarrow{}{} P2)
Date html generated:
2016_05_14-AM-06_05_47
Last ObjectModification:
2015_12_26-AM-11_32_21
Theory : relations
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