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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 7416 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶)) | |
2 | oveq1 7416 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶)) | |
3 | 1, 2 | ifsb 4542 | 1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ifcif 4529 (class class class)co 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: scmatscm 22015 pmatcollpwscmatlem1 22291 idpm2idmp 22303 monmat2matmon 22326 chmatval 22331 leibpi 26447 musumsum 26696 muinv 26697 dchrinvcl 26756 rpvmasum2 27015 padicabvcxp 27135 pnfneige0 32931 plymulx0 33558 ftc1anclem6 36566 reabssgn 42387 sqrtcval 42392 linc0scn0 47104 |
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