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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 1049 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | |
2 | iftrue 4496 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | iftrue 4496 | . . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
4 | 3 | eqcomd 2739 | . . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵)) |
5 | 2, 4 | sylan9eq 2793 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
6 | iffalse 4499 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
7 | iffalse 4499 | . . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2739 | . . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵)) |
9 | 6, 8 | sylan9eq 2793 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
10 | 5, 9 | jaoi 856 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
11 | 1, 10 | sylbi 216 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ifcif 4490 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-if 4491 |
This theorem is referenced by: ifbid 4513 ifbieq2i 4515 gsummoncoe1 21698 scmatscm 21885 mulmarep1gsum1 21945 madugsum 22015 mp2pm2mplem4 22181 dchrhash 26642 lgsdi 26705 rpvmasum2 26883 ifnebib 31521 bj-projval 35517 matunitlindflem2 36125 itg2gt0cn 36183 dedths 37474 dfafv2 45454 |
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