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Theorem ifbieq2i 3463
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 3460 . . 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 3446 . . 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 2136 1 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  ifcif 3442
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-un 3043  df-if 3443
This theorem is referenced by:  ifbieq12i  3465  gcdcom  11558  gcdass  11599  lcmcom  11641  lcmass  11662
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