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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 6500 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | fveq1 6500 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐺‘𝐴)) | |
3 | 1, 2 | ifsb 4364 | 1 ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ifcif 4351 ‘cfv 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-rex 3094 df-if 4352 df-uni 4714 df-br 4931 df-iota 6154 df-fv 6198 |
This theorem is referenced by: decpmatid 21085 pmatcollpwscmatlem1 21104 |
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