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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 4531 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
| 2 | iftrue 4531 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶) | |
| 3 | 1, 2 | oveq12d 7449 | . . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶)) |
| 4 | iftrue 4531 | . . 3 ⊢ (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶)) | |
| 5 | 3, 4 | eqtr4d 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) | |
| 8 | 6, 7 | oveq12d 7449 | . . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷)) |
| 9 | iffalse 4534 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷)) | |
| 10 | 8, 9 | eqtr4d 2780 | . 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 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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