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Theorem ifbieq2d 3415
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 3412 . 2 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴))
3 ifbieq2d.2 . . 3 (𝜑𝐴 = 𝐵)
43ifeq2d 3409 . 2 (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
52, 4eqtrd 2120 1 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  ifcif 3393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3003  df-if 3394
This theorem is referenced by:  xnegeq  9289  iseqf1olemqval  9916  iseqf1olemqk  9923  seq3f1olemqsum  9929  exp3val  9957  gcdval  11229  gcdass  11282  lcmval  11323  lcmass  11345  nnsf  11895  peano4nninf  11896  peano3nninf  11897  exmidsbthr  11913
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