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| 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 4530 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
| 3 | iftrue 4530 | . . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
| 4 | 3 | eqcomd 2742 | . . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵)) | 
| 5 | 2, 4 | sylan9eq 2796 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | 
| 6 | iffalse 4533 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
| 7 | iffalse 4533 | . . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
| 8 | 7 | eqcomd 2742 | . . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵)) | 
| 9 | 6, 8 | sylan9eq 2796 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | 
| 10 | 5, 9 | jaoi 857 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | 
| 11 | 1, 10 | sylbi 217 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ifcif 4524 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-if 4525 | 
| This theorem is referenced by: ifbid 4548 ifbieq2i 4550 prodeq1i 15953 psdmvr 22174 gsummoncoe1 22313 scmatscm 22520 mulmarep1gsum1 22580 madugsum 22650 mp2pm2mplem4 22816 dchrhash 27316 lgsdi 27379 rpvmasum2 27557 ifnebib 32563 itgeq12i 36208 bj-projval 36998 matunitlindflem2 37625 itg2gt0cn 37683 dedths 38964 dfafv2 47149 | 
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