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Theorem ifbieq1d 3413
Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
Hypotheses
Ref Expression
ifbieq1d.1 (𝜑 → (𝜓𝜒))
ifbieq1d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ifbieq1d (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))

Proof of Theorem ifbieq1d
StepHypRef Expression
1 ifbieq1d.1 . . 3 (𝜑 → (𝜓𝜒))
21ifbid 3412 . 2 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐴, 𝐶))
3 ifbieq1d.2 . . 3 (𝜑𝐴 = 𝐵)
43ifeq1d 3408 . 2 (𝜑 → if(𝜒, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
52, 4eqtrd 2120 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  ifcif 3393
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3003  df-if 3394
This theorem is referenced by:  iseqf1olemfvp  9926  seq3f1olemqsum  9929  seq3f1oleml  9932  seq3f1o  9933  bcval  10157  isumrblem  10765  isummolem3  10770  isummolem2a  10771  isummo  10773  zisum  10774  fisum  10778  fsum3  10779  isumss  10783  isumss2  10785  fisumcvg2  10786  fsum3cvg2  10787  fisumser  10790  fsumcl2lem  10792  fsumadd  10800  sumsnf  10803  fsummulc2  10842  isumlessdc  10890
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