Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elim2if | Structured version Visualization version GIF version |
Description: Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
elim2if.1 | ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃)) |
elim2if.2 | ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏)) |
elim2if.3 | ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂)) |
Ref | Expression |
---|---|
elim2if | ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4042 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴) | |
2 | elim2if.1 | . . 3 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
4 | iffalse 4045 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, 𝐶)) | |
5 | 4 | eqeq1d 2612 | . . . 4 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 ↔ if(𝜓, 𝐵, 𝐶) = 𝐵)) |
6 | elim2if.2 | . . . 4 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏)) | |
7 | 5, 6 | syl6bir 243 | . . 3 ⊢ (¬ 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐵 → (𝜒 ↔ 𝜏))) |
8 | 4 | eqeq1d 2612 | . . . 4 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 ↔ if(𝜓, 𝐵, 𝐶) = 𝐶)) |
9 | elim2if.3 | . . . 4 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂)) | |
10 | 8, 9 | syl6bir 243 | . . 3 ⊢ (¬ 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐶 → (𝜒 ↔ 𝜂))) |
11 | 7, 10 | elimifd 28746 | . 2 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))) |
12 | 3, 11 | cases 1004 | 1 ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = 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: elim2ifim 28748 |
Copyright terms: Public domain | W3C validator |