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: implies:  Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  predicate_equivalent: P1 ⇐⇒ P2 uall: [x:A]. B[x] implies:  Q all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q member: t ∈ T prop: rev_implies:  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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