### Nuprl Lemma : ifthenelse_functionality_wrt_implies2

`∀b1,b2:𝔹.  ∀[p,q1,q2:ℙ].  (b1 = b2 `` {q1 `` q2} `` {if b1 then p else q1 fi  `` if b2 then p 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]` 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` 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` rev_implies: `P `` Q` sq_type: `SQType(T)` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` bnot: `¬bb` false: `False`
Lemmas referenced :  bool_wf 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_cases_sqequal assert_of_bnot ifthenelse_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation isect_memberFormation cut hypothesisEquality thin introduction extract_by_obid hypothesis sqequalHypSubstitution unionElimination equalityElimination isectElimination productElimination independent_isectElimination instantiate cumulativity independent_pairFormation natural_numberEquality because_Cache dependent_functionElimination independent_functionElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp voidElimination universeEquality functionEquality

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

Date html generated: 2017_04_14-AM-07_30_03
Last ObjectModification: 2017_02_27-PM-02_58_35

Theory : bool_1

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