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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 1044 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | |
2 | iftrue 4476 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | iftrue 4476 | . . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
4 | 3 | eqcomd 2830 | . . . 4 ⊢ (𝜓 → 𝐴 = if(𝜓, 𝐴, 𝐵)) |
5 | 2, 4 | sylan9eq 2879 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
6 | iffalse 4479 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
7 | iffalse 4479 | . . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2830 | . . . 4 ⊢ (¬ 𝜓 → 𝐵 = if(𝜓, 𝐴, 𝐵)) |
9 | 6, 8 | sylan9eq 2879 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
10 | 5, 9 | jaoi 853 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
11 | 1, 10 | sylbi 219 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ifcif 4470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-if 4471 |
This theorem is referenced by: ifbid 4492 ifbieq2i 4494 gsummoncoe1 20475 scmatscm 21125 mulmarep1gsum1 21185 madugsum 21255 mp2pm2mplem4 21420 dchrhash 25850 lgsdi 25913 rpvmasum2 26091 bj-projval 34312 matunitlindflem2 34893 itg2gt0cn 34951 dedths 36102 dfafv2 43338 |
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