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Theorem ifeq12 4247
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 4234 . 2 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2 ifeq2 4235 . 2 (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
31, 2sylan9eq 2814 1 ((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  ifcif 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-un 3720  df-if 4231
This theorem is referenced by:  xaddmnf1  12272  xpslem  16455  ditg0  23836  mumullem2  25126
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