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Theorem ovif12 7533
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 4531 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 iftrue 4531 . . . 4 (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
31, 2oveq12d 7449 . . 3 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4 iftrue 4531 . . 3 (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
53, 4eqtr4d 2780 . 2 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6 iffalse 4534 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4534 . . . 4 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
86, 7oveq12d 7449 . . 3 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9 iffalse 4534 . . 3 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
108, 9eqtr4d 2780 . 2 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
115, 10pm2.61i 182 1 (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  ifcif 4525  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  ofccat  15008  limccnp2  25927  ftc1anclem5  37704  fsuppind  42600  sqrtcval  43654
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