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Theorem ifeq12 4539
Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
Assertion
Ref Expression
ifeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Proof of Theorem ifeq12
StepHypRef Expression
1 ifeq1 4525 . 2 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2 ifeq2 4526 . 2 (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
31, 2sylan9eq 2784 1 ((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  ifcif 4521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-un 3946  df-if 4522
This theorem is referenced by:  xaddmnf1  13208  xpsrnbas  17522  ditg0  25726  mumullem2  27053  sqrtcval  42942  sqrtcval2  42943
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