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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 6895 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | fveq1 6895 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐺‘𝐴)) | |
3 | 1, 2 | ifsb 4543 | 1 ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ifcif 4530 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-ss 3961 df-if 4531 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 |
This theorem is referenced by: decpmatid 22716 pmatcollpwscmatlem1 22735 selvvvval 41953 prjspnfv01 42183 |
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