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Theorem ifbieq12d 3572
Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
ifbieq12d.1 (𝜑 → (𝜓𝜒))
ifbieq12d.2 (𝜑𝐴 = 𝐶)
ifbieq12d.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
ifbieq12d (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Proof of Theorem ifbieq12d
StepHypRef Expression
1 ifbieq12d.1 . . 3 (𝜑 → (𝜓𝜒))
21ifbid 3567 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
3 ifbieq12d.2 . . 3 (𝜑𝐴 = 𝐶)
4 ifbieq12d.3 . . 3 (𝜑𝐵 = 𝐷)
53, 4ifeq12d 3565 . 2 (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
62, 5eqtrd 2220 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  ifcif 3546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-un 3145  df-if 3547
This theorem is referenced by:  updjudhcoinlf  7093  updjudhcoinrg  7094  omp1eom  7108  xaddval  9859  iseqf1olemqval  10501  iseqf1olemqk  10508  seq3f1olemqsum  10514  exp3val  10536  cvgratz  11554  eucalgval2  12067  ennnfonelemg  12418  ennnfonelem1  12422  mulgval  13017  lgsval  14701
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