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Theorem ifeq1dadc 3401
Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
Hypotheses
Ref Expression
ifeq1dadc.1 ((𝜑𝜓) → 𝐴 = 𝐵)
ifeq1dadc.dc (𝜑DECID 𝜓)
Assertion
Ref Expression
ifeq1dadc (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1dadc
StepHypRef Expression
1 ifeq1dadc.1 . . 3 ((𝜑𝜓) → 𝐴 = 𝐵)
21ifeq1d 3388 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3 iffalse 3381 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4 iffalse 3381 . . . 4 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
53, 4eqtr4d 2118 . . 3 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
65adantl 271 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7 ifeq1dadc.dc . . 3 (𝜑DECID 𝜓)
8 exmiddc 778 . . 3 (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
97, 8syl 14 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
102, 6, 9mpjaodan 745 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 776   = wceq 1285  ifcif 3373
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-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-un 2988  df-if 3374
This theorem is referenced by:  sumeq2d  10568  sumeq2  10569
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