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Mirrors > Home > MPE Home > Th. List > Mathboxes > elimifd | Structured version Visualization version GIF version |
Description: Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
elimifd.1 | ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒 ↔ 𝜃))) |
elimifd.2 | ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒 ↔ 𝜏))) |
Ref | Expression |
---|---|
elimifd | ⊢ (𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜃) ∨ (¬ 𝜓 ∧ 𝜏)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid 888 | . . . 4 ⊢ (𝜓 ∨ ¬ 𝜓) | |
2 | 1 | biantrur 531 | . . 3 ⊢ (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒)) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒))) |
4 | andir 1002 | . . 3 ⊢ (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜒))) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜒)))) |
6 | iftrue 4469 | . . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
7 | elimifd.1 | . . . . 5 ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒 ↔ 𝜃))) | |
8 | 6, 7 | syl5 34 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
9 | 8 | pm5.32d 577 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) |
10 | iffalse 4472 | . . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
11 | elimifd.2 | . . . . 5 ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒 ↔ 𝜏))) | |
12 | 10, 11 | syl5 34 | . . . 4 ⊢ (𝜑 → (¬ 𝜓 → (𝜒 ↔ 𝜏))) |
13 | 12 | pm5.32d 577 | . . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ 𝜒) ↔ (¬ 𝜓 ∧ 𝜏))) |
14 | 9, 13 | orbi12d 912 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜃) ∨ (¬ 𝜓 ∧ 𝜏)))) |
15 | 3, 5, 14 | 3bitrd 306 | 1 ⊢ (𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜃) ∨ (¬ 𝜓 ∧ 𝜏)))) |
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: elim2if 30226 |
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