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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 7141 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵)) | |
2 | oveq 7141 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵)) | |
3 | 1, 2 | ifsb 4438 | 1 ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ifcif 4425 (class class class)co 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-if 4426 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: monmatcollpw 21384 |
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