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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 6669 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐹‘𝐴)) | |
2 | fveq1 6669 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (if(𝜑, 𝐹, 𝐺)‘𝐴) = (𝐺‘𝐴)) | |
3 | 1, 2 | ifsb 4480 | 1 ⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ifcif 4467 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-if 4468 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 |
This theorem is referenced by: decpmatid 21378 pmatcollpwscmatlem1 21397 |
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