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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 6102 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | fveq1 6102 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐺‘𝐴)) | |
3 | 1, 2 | ifsb 4049 | 1 ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ifcif 4036 ‘cfv 5804 |
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-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-if 4037 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: decpmatid 20394 pmatcollpwscmatlem1 20413 |
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