Nuprl Lemma : predicate_equivalent_weakening

[T:Type]. ∀[P1,P2:T ⟶ ℙ].  P1 ⇐⇒ P2 supposing P1 P2 ∈ (T ⟶ ℙ)


Proof




Definitions occuring in Statement :  predicate_equivalent: P1 ⇐⇒ P2 uimplies: supposing a uall: [x:A]. B[x] prop: function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  predicate_equivalent: P1 ⇐⇒ P2 uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q implies:  Q subtype_rel: A ⊆B prop: rev_implies:  Q
Lemmas referenced :  and_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation cut introduction axiomEquality hypothesis thin rename lambdaFormation independent_pairFormation dependent_set_memberEquality hypothesisEquality instantiate extract_by_obid sqequalHypSubstitution isectElimination functionEquality applyEquality lambdaEquality cumulativity universeEquality because_Cache setElimination productElimination setEquality equalitySymmetry hyp_replacement Error :applyLambdaEquality,  functionExtensionality

Latex:
\mforall{}[T:Type].  \mforall{}[P1,P2:T  {}\mrightarrow{}  \mBbbP{}].    P1  \mLeftarrow{}{}\mRightarrow{}  P2  supposing  P1  =  P2



Date html generated: 2016_10_21-AM-09_43_35
Last ObjectModification: 2016_07_12-AM-05_04_11

Theory : relations


Home Index