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| Mirrors > Home > MPE Home > Th. List > iffv | Structured version Visualization version GIF version | ||
| Description: Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| iffv | ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq1 6905 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
| 2 | fveq1 6905 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐺‘𝐴)) | |
| 3 | 1, 2 | ifsb 4539 | 1 ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) | 
| 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-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 | 
| This theorem is referenced by: decpmatid 22776 pmatcollpwscmatlem1 22795 selvvvval 42595 prjspnfv01 42634 | 
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