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Theorem ifeq12 4466
 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 4453 . 2 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2 ifeq2 4454 . 2 (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
31, 2sylan9eq 2879 1 ((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ifcif 4449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-if 4450 This theorem is referenced by:  xaddmnf1  12614  xpsrnbas  16840  ditg0  24452  mumullem2  25761  sqrtcval  40194  sqrtcval2  40195
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