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Theorem ifbieq2i 4474
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 4471 . . 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 4455 . . 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 2847 1 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  ifcif 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-if 4451
This theorem is referenced by:  ifbieq12i  4476  gcdcom  15860  gcdass  15893  lcmcom  15935  lcmass  15956  bj-xpimasn  34335  cdleme31sdnN  37628  cdlemefr44  37666  cdleme48fv  37740  cdlemeg49lebilem  37780  cdleme50eq  37782  hoidmvlelem3  43162  hoidmvlelem4  43163
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