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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 6555 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵)) | |
2 | oveq 6555 | . 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵)) | |
3 | 1, 2 | ifsb 4049 | 1 ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ifcif 4036 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-if 4037 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: monmatcollpw 20403 |
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