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Theorem ovif12 7456
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 4492 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 iftrue 4492 . . . 4 (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
31, 2oveq12d 7375 . . 3 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4 iftrue 4492 . . 3 (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
53, 4eqtr4d 2779 . 2 (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6 iffalse 4495 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4495 . . . 4 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
86, 7oveq12d 7375 . . 3 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9 iffalse 4495 . . 3 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
108, 9eqtr4d 2779 . 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 1541  ifcif 4486  (class class class)co 7357
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360
This theorem is referenced by:  ofccat  14854  limccnp2  25256  ftc1anclem5  36155  fsuppind  40751  sqrtcval  41903
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