### Nuprl Lemma : ifthenelse_functionality_wrt_rev_implies

`∀b1,b2:𝔹.`
`  ∀[p1,q1,p2,q2:ℙ].  (b1 = b2 `` {q1 `` q2} `` {p1 `` p2} `` {if b1 then p1 else q1 fi  `` if b2 then p2 else q2 fi })`

Proof

Definitions occuring in Statement :  ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` all: `∀x:A. B[x]` rev_implies: `P `` Q` implies: `P `` Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  guard: `{T}` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` rev_implies: `P `` Q` member: `t ∈ T` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` true: `True` prop: `ℙ` sq_type: `SQType(T)` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` bnot: `¬bb` false: `False`
Lemmas referenced :  eqtt_to_assert subtype_base_sq bool_subtype_base iff_imp_equal_bool btrue_wf assert_wf true_wf eqff_to_assert equal_wf bool_wf bool_cases_sqequal assert_of_bnot ifthenelse_wf rev_implies_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation isect_memberFormation cut hypothesisEquality thin because_Cache sqequalHypSubstitution unionElimination equalityElimination introduction extract_by_obid isectElimination hypothesis productElimination independent_isectElimination independent_functionElimination instantiate independent_pairFormation natural_numberEquality equalitySymmetry dependent_functionElimination equalityTransitivity dependent_pairFormation promote_hyp voidElimination cumulativity universeEquality

Latex:
\mforall{}b1,b2:\mBbbB{}.
\mforall{}[p1,q1,p2,q2:\mBbbP{}].
(b1  =  b2
{}\mRightarrow{}  \{q1  \mLeftarrow{}{}  q2\}
{}\mRightarrow{}  \{p1  \mLeftarrow{}{}  p2\}
{}\mRightarrow{}  \{if  b1  then  p1  else  q1  fi    \mLeftarrow{}{}  if  b2  then  p2  else  q2  fi  \})

Date html generated: 2017_04_14-AM-07_29_59
Last ObjectModification: 2017_02_27-PM-02_58_36

Theory : bool_1

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