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Theorem ifeq1d 3542
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ifeq1d (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1d
StepHypRef Expression
1 ifeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ifeq1 3528 . 2 (𝐴 = 𝐵 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
31, 2syl 14 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  ifcif 3525
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-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 3526
This theorem is referenced by:  ifeq12d  3544  ifbieq1d  3547  ifeq1dadc  3555  iseqf1olemjpcl  10440  iseqf1olemqpcl  10441  iseqf1olemfvp  10442  seq3f1olemqsum  10445  seq3f1olemp  10447  summodc  11335  fsum3  11339  fsum3ser  11349  isumlessdc  11448  prodeq2w  11508  prodmodc  11530  fprodseq  11535  prodssdc  11541  lgsval  13660
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