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| Mirrors > Home > MPE Home > Th. List > fvif | Structured version Visualization version GIF version | ||
| Description: Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| fvif | ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴)) | |
| 2 | fveq2 6906 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵)) | |
| 3 | 1, 2 | ifsb 4539 | 1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ifcif 4525 ‘cfv 6561 |
| 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 |
| This theorem is referenced by: ccatco 14874 sumeq2ii 15729 prodeq2ii 15947 ruclem1 16267 xpsrnbas 17616 rhmmpl 22387 rhmply1vr1 22391 mat2pmat1 22738 decpmatid 22776 pmatcollpwscmatlem1 22795 copco 25051 pcopt 25055 pcopt2 25056 limccnp 25926 prmorcht 27221 pclogsum 27259 mblfinlem2 37665 ftc1anclem8 37707 ftc1anc 37708 rhmpsr 42562 fvifeq 47292 |
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