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Theorem ifbieq2i 3549
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 3546 . . 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 3530 . . 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 2191 1 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1348  ifcif 3526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3527
This theorem is referenced by:  ifbieq12i  3551  gcdcom  11928  gcdass  11970  lcmcom  12018  lcmass  12039
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