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Theorem ifbieq2i 4484
Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2i.1 (𝜑𝜓)
ifbieq2i.2 𝐴 = 𝐵
Assertion
Ref Expression
ifbieq2i if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Proof of Theorem ifbieq2i
StepHypRef Expression
1 ifbieq2i.1 . . 3 (𝜑𝜓)
2 ifbi 4481 . . 3 ((𝜑𝜓) → if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴))
31, 2ax-mp 5 . 2 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴)
4 ifbieq2i.2 . . 3 𝐴 = 𝐵
5 ifeq2 4464 . . 3 (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
64, 5ax-mp 5 . 2 if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
73, 6eqtri 2766 1 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  ifcif 4459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-un 3891  df-if 4460
This theorem is referenced by:  ifbieq12i  4486  gcdcom  16230  gcdass  16265  lcmcom  16308  lcmass  16329  bj-xpimasn  35153  cdleme31sdnN  38409  cdlemefr44  38447  cdleme48fv  38521  cdlemeg49lebilem  38561  cdleme50eq  38563  hoidmvlelem3  44116  hoidmvlelem4  44117
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