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Theorem ifbieq2d 3491
Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2d.1 (𝜑 → (𝜓𝜒))
ifbieq2d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ifbieq2d (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Proof of Theorem ifbieq2d
StepHypRef Expression
1 ifbieq2d.1 . . 3 (𝜑 → (𝜓𝜒))
21ifbid 3488 . 2 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴))
3 ifbieq2d.2 . . 3 (𝜑𝐴 = 𝐵)
43ifeq2d 3485 . 2 (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
52, 4eqtrd 2170 1 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  ifcif 3469
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-if 3470
This theorem is referenced by:  difinfsnlem  6977  ctmlemr  6986  xnegeq  9603  xaddval  9621  iseqf1olemqval  10253  iseqf1olemqk  10260  seq3f1olemqsum  10266  exp3val  10288  gcdval  11637  gcdass  11692  lcmval  11733  lcmass  11755  ennnfonelemj0  11903  ennnfonelemjn  11904  ennnfonelem0  11907  ennnfonelemp1  11908  ennnfonelemnn0  11924  nnsf  13188  peano4nninf  13189  peano3nninf  13190  exmidsbthr  13207
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