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| Mirrors > Home > MPE Home > Th. List > ifov | Structured version Visualization version GIF version | ||
| Description: Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.) |
| Ref | Expression |
|---|---|
| ifov | ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq 7437 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵)) | |
| 2 | oveq 7437 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵)) | |
| 3 | 1, 2 | ifsb 4539 | 1 ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: = 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-if 4526 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: monmatcollpw 22785 |
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