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Mirrors > Home > MPE Home > Th. List > ovif | Structured version Visualization version GIF version |
Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
ovif | ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6556 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶)) | |
2 | oveq1 6556 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶)) | |
3 | 1, 2 | ifsb 4049 | 1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ifcif 4036 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: scmatscm 20138 pmatcollpwscmatlem1 20413 idpm2idmp 20425 monmat2matmon 20448 chmatval 20453 leibpi 24469 musumsum 24718 muinv 24719 dchrinvcl 24778 rpvmasum2 25001 padicabvcxp 25121 pnfneige0 29325 plymulx0 29950 ftc1anclem6 32660 linc0scn0 42006 |
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