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Mirrors > Home > MPE Home > Th. List > ovif12 | Structured version Visualization version GIF version |
Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
ovif12 | ⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4492 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | iftrue 4492 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶) | |
3 | 1, 2 | oveq12d 7375 | . . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶)) |
4 | iftrue 4492 | . . 3 ⊢ (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶)) | |
5 | 3, 4 | eqtr4d 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) | |
8 | 6, 7 | oveq12d 7375 | . . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷)) |
9 | iffalse 4495 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷)) | |
10 | 8, 9 | eqtr4d 2779 | . 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))) |
11 | 5, 10 | pm2.61i 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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