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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 4092 | . . 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 4095 | . . 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 992 | 1 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: eqif 4126 elif 4128 ifel 4129 ftc1anclem5 33489 clsk1indlem2 38340 |
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