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Theorem ifeq12d 3414
Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
Hypotheses
Ref Expression
ifeq1d.1 (𝜑𝐴 = 𝐵)
ifeq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
ifeq12d (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Proof of Theorem ifeq12d
StepHypRef Expression
1 ifeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21ifeq1d 3412 . 2 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3 ifeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43ifeq2d 3413 . 2 (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷))
52, 4eqtrd 2121 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  ifcif 3397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-if 3398
This theorem is referenced by:  ifbieq12d  3421  exp3val  10011  eucalgval  11368
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