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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 6446 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴)) | |
2 | fveq2 6446 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵)) | |
3 | 1, 2 | ifsb 4320 | 1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ifcif 4307 ‘cfv 6135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 |
This theorem is referenced by: ccatco 13986 sumeq2ii 14831 prodeq2ii 15046 ruclem1 15364 xpslem 16619 mat2pmat1 20944 decpmatid 20982 pmatcollpwscmatlem1 21001 copco 23225 pcopt 23229 pcopt2 23230 limccnp 24092 prmorcht 25356 pclogsum 25392 mblfinlem2 34073 ftc1anclem8 34117 ftc1anc 34118 fvifeq 42321 |
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