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Mirrors > Home > MPE Home > Th. List > elimdhyp | Structured version Visualization version GIF version |
Description: Version of elimhyp 4526 where the hypothesis is deduced from the final antecedent. See divalg 15742 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.) |
Ref | Expression |
---|---|
elimdhyp.1 | ⊢ (𝜑 → 𝜓) |
elimdhyp.2 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) |
elimdhyp.3 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒)) |
elimdhyp.4 | ⊢ 𝜃 |
Ref | Expression |
---|---|
elimdhyp | ⊢ 𝜒 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimdhyp.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | iftrue 4469 | . . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | 2 | eqcomd 2824 | . . . 4 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵)) |
4 | elimdhyp.2 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
6 | 1, 5 | mpbid 233 | . 2 ⊢ (𝜑 → 𝜒) |
7 | elimdhyp.4 | . . 3 ⊢ 𝜃 | |
8 | iffalse 4472 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
9 | 8 | eqcomd 2824 | . . . 4 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵)) |
10 | elimdhyp.3 | . . . 4 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (¬ 𝜑 → (𝜃 ↔ 𝜒)) |
12 | 7, 11 | mpbii 234 | . 2 ⊢ (¬ 𝜑 → 𝜒) |
13 | 6, 12 | pm2.61i 183 | 1 ⊢ 𝜒 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = 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: divalg 15742 |
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