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Theorem ovif12 7252
 Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
ovif12 (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))

Proof of Theorem ovif12
StepHypRef Expression
1 iftrue 4429 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 iftrue 4429 . . . 4 (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
31, 2oveq12d 7173 . . 3 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4 iftrue 4429 . . 3 (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
53, 4eqtr4d 2796 . 2 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6 iffalse 4432 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4432 . . . 4 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
86, 7oveq12d 7173 . . 3 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9 iffalse 4432 . . 3 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
108, 9eqtr4d 2796 . 2 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
115, 10pm2.61i 185 1 (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538  ifcif 4423  (class class class)co 7155 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347  df-ov 7158 This theorem is referenced by:  ofccat  14381  limccnp2  24596  ftc1anclem5  35440  fsuppind  39812  sqrtcval  40742
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