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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 6673 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴)) | |
2 | fveq2 6673 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵)) | |
3 | 1, 2 | ifsb 4483 | 1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ifcif 4470 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 |
This theorem is referenced by: ccatco 14200 sumeq2ii 15053 prodeq2ii 15270 ruclem1 15587 xpsrnbas 16847 mat2pmat1 21343 decpmatid 21381 pmatcollpwscmatlem1 21400 copco 23625 pcopt 23629 pcopt2 23630 limccnp 24492 prmorcht 25758 pclogsum 25794 mblfinlem2 34934 ftc1anclem8 34978 ftc1anc 34979 fvifeq 43486 |
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