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Mirrors > Home > MPE Home > Th. List > ifov | Structured version Visualization version GIF version |
Description: Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.) |
Ref | Expression |
---|---|
ifov | ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7281 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵)) | |
2 | oveq 7281 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵)) | |
3 | 1, 2 | ifsb 4472 | 1 ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ifcif 4459 (class class class)co 7275 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-in 3894 df-ss 3904 df-if 4460 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 |
This theorem is referenced by: monmatcollpw 21928 |
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