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Theorem ifbieq12d 3354
 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 3349 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
3 ifbieq12d.2 . . 3 (𝜑𝐴 = 𝐶)
4 ifbieq12d.3 . . 3 (𝜑𝐵 = 𝐷)
53, 4ifeq12d 3347 . 2 (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
62, 5eqtrd 2072 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = 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-in1 544  ax-in2 545  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:  expival  9257
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