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Theorem ifeq1d 3345
 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 3334 . 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 1243  ifcif 3331 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-if 3332 This theorem is referenced by:  ifeq12d  3347  ifbieq1d  3350
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