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Mirrors > Home > MPE Home > Th. List > fvif | Structured version Visualization version GIF version |
Description: Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
fvif | ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6843 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴)) | |
2 | fveq2 6843 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵)) | |
3 | 1, 2 | ifsb 4500 | 1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ifcif 4487 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 |
This theorem is referenced by: ccatco 14725 sumeq2ii 15579 prodeq2ii 15797 ruclem1 16114 xpsrnbas 17454 mat2pmat1 22084 decpmatid 22122 pmatcollpwscmatlem1 22141 copco 24384 pcopt 24388 pcopt2 24389 limccnp 25258 prmorcht 26530 pclogsum 26566 mblfinlem2 36119 ftc1anclem8 36161 ftc1anc 36162 rhmmpl 40744 fvifeq 45519 |
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