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Theorem ifeq1dadc 3555
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 3542 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3 iffalse 3533 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4 iffalse 3533 . . . 4 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
53, 4eqtr4d 2206 . . 3 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
65adantl 275 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7 ifeq1dadc.dc . . 3 (𝜑DECID 𝜓)
8 exmiddc 831 . . 3 (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
97, 8syl 14 . 2 (𝜑 → (𝜓 ∨ ¬ 𝜓))
102, 6, 9mpjaodan 793 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 703  DECID wdc 829   = wceq 1348  ifcif 3525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3526
This theorem is referenced by:  sumeq2  11309  isumss  11341  prodeq2  11507  lgsval2lem  13626  lgsval4lem  13627  lgsneg  13640  lgsmod  13642  lgsdilem2  13652
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