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Mirrors > Home > MPE Home > Th. List > ifbi | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Ref | Expression |
---|---|
ifbi | ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3 1041 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | |
2 | iftrue 4469 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | iftrue 4469 | . . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
4 | 3 | eqcomd 2824 | . . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵)) |
5 | 2, 4 | sylan9eq 2873 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
6 | iffalse 4472 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
7 | iffalse 4472 | . . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2824 | . . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵)) |
9 | 6, 8 | sylan9eq 2873 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
10 | 5, 9 | jaoi 851 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
11 | 1, 10 | sylbi 218 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-if 4464 |
This theorem is referenced by: ifbid 4485 ifbieq2i 4487 gsummoncoe1 20400 scmatscm 21050 mulmarep1gsum1 21110 madugsum 21180 mp2pm2mplem4 21345 dchrhash 25774 lgsdi 25837 rpvmasum2 26015 bj-projval 34205 matunitlindflem2 34770 itg2gt0cn 34828 dedths 35978 dfafv2 43208 |
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