Nuprl Lemma : predicate_equivalent_transitivity
∀[T:Type]. ∀[P1,P2,P3:T ⟶ ℙ]. (P1 ⇐⇒ P2 ⇒ P2 ⇐⇒ P3 ⇒ P1 ⇐⇒ P3)
Proof
Definitions occuring in Statement :
predicate_equivalent: P1 ⇐⇒ P2,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
predicate_equivalent: P1 ⇐⇒ P2,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
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,P3:T {}\mrightarrow{} \mBbbP{}]. (P1 \mLeftarrow{}{}\mRightarrow{} P2 {}\mRightarrow{} P2 \mLeftarrow{}{}\mRightarrow{} P3 {}\mRightarrow{} P1 \mLeftarrow{}{}\mRightarrow{} P3)
Date html generated:
2016_05_14-AM-06_05_55
Last ObjectModification:
2015_12_26-AM-11_32_28
Theory : relations
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