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Mirrors > Home > MPE Home > Th. List > ifor | Structured version Visualization version GIF version |
Description: Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
ifor | ⊢ if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4042 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = 𝐴) | |
2 | 1 | orcs 408 | . . 3 ⊢ (𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = 𝐴) |
3 | iftrue 4042 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) = 𝐴) | |
4 | 2, 3 | eqtr4d 2647 | . 2 ⊢ (𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))) |
5 | iffalse 4045 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) = if(𝜓, 𝐴, 𝐵)) | |
6 | biorf 419 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
7 | 6 | ifbid 4058 | . . 3 ⊢ (¬ 𝜑 → if(𝜓, 𝐴, 𝐵) = if((𝜑 ∨ 𝜓), 𝐴, 𝐵)) |
8 | 5, 7 | eqtr2d 2645 | . 2 ⊢ (¬ 𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))) |
9 | 4, 8 | pm2.61i 175 | 1 ⊢ if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 = wceq 1475 ifcif 4036 |
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-if 4037 |
This theorem is referenced by: cantnflem1d 8468 cantnflem1 8469 |
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