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Mirrors > Home > MPE Home > Th. List > elimif | Structured version Visualization version GIF version |
Description: Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.) |
Ref | Expression |
---|---|
elimif.1 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓 ↔ 𝜒)) |
elimif.2 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
elimif | ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4042 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | elimif.1 | . . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
4 | iffalse 4045 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
5 | elimif.2 | . . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃)) |
7 | 3, 6 | cases 1004 | 1 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) |
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: eqif 4076 elif 4078 ifel 4079 ftc1anclem5 32659 clsk1indlem2 37360 |
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