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Theorem ifeq12d 3486
 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 3484 . 2 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3 ifeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43ifeq2d 3485 . 2 (𝜑 → if(𝜓, 𝐵, 𝐶) = if(𝜓, 𝐵, 𝐷))
52, 4eqtrd 2170 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ifcif 3469 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-if 3470 This theorem is referenced by:  ifbieq12d  3493  xaddpnf1  9622  exp3val  10288  eucalgval  11724  ennnfonelemp1  11908  ennnfonelemnn0  11924
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