Nuprl Lemma : ab_binrel_functionality

`∀[T:Type]. ∀[E,E':T ⟶ T ⟶ ℙ].  ((∀x,y:T.  (E[x;y] `⇐⇒` E'[x;y])) `` ((x,y:T. E[x;y]) <≡>{T} (x,y:T. E'[x;y])))`

Proof

Definitions occuring in Statement :  ab_binrel: `x,y:T. E[x; y]` binrel_eqv: `E <≡>{T} E'` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` ab_binrel: `x,y:T. E[x; y]` binrel_eqv: `E <≡>{T} E'` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` so_apply: `x[s]`
Lemmas referenced :  all_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule hypothesis cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality functionEquality cumulativity universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[E,E':T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}x,y:T.    (E[x;y]  \mLeftarrow{}{}\mRightarrow{}  E'[x;y]))  {}\mRightarrow{}  ((x,y:T.  E[x;y])  <\mequiv{}>\{T\}  (x,y:T.  E'[x;y])))

Date html generated: 2016_05_15-PM-00_00_36
Last ObjectModification: 2015_12_26-PM-11_26_39

Theory : gen_algebra_1

Home Index