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Theorem ifeq2dadc 3577
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
ifeq2da.1 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
ifeq2dadc.dc (𝜑DECID 𝜓)
Assertion
Ref Expression
ifeq2dadc (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2dadc
StepHypRef Expression
1 simpr 110 . . . 4 ((𝜑𝜓) → 𝜓)
21iftrued 3553 . . 3 ((𝜑𝜓) → if(𝜓, 𝐶, 𝐴) = 𝐶)
31iftrued 3553 . . 3 ((𝜑𝜓) → if(𝜓, 𝐶, 𝐵) = 𝐶)
42, 3eqtr4d 2223 . 2 ((𝜑𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5 ifeq2da.1 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
65ifeq2d 3564 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7 ifeq2dadc.dc . . 3 (𝜑DECID 𝜓)
8 exmiddc 837 . . 3 (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
97, 8syl 14 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
104, 6, 9mpjaodan 799 1 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  DECID wdc 835   = 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-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-dc 836  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:  subgmulg  13077
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