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Mirrors > Home > MPE Home > Th. List > keephyp2v | Structured version Visualization version GIF version |
Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4089). (Contributed by NM, 16-Apr-2005.) |
Ref | Expression |
---|---|
keephyp2v.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) |
keephyp2v.2 | ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) |
keephyp2v.3 | ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂)) |
keephyp2v.4 | ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃)) |
keephyp2v.5 | ⊢ 𝜓 |
keephyp2v.6 | ⊢ 𝜏 |
Ref | Expression |
---|---|
keephyp2v | ⊢ 𝜃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | keephyp2v.5 | . . 3 ⊢ 𝜓 | |
2 | iftrue 4042 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐴) | |
3 | 2 | eqcomd 2616 | . . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐶)) |
4 | keephyp2v.1 | . . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
6 | iftrue 4042 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐵) | |
7 | 6 | eqcomd 2616 | . . . . 5 ⊢ (𝜑 → 𝐵 = if(𝜑, 𝐵, 𝐷)) |
8 | keephyp2v.2 | . . . . 5 ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
10 | 5, 9 | bitrd 267 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
11 | 1, 10 | mpbii 222 | . 2 ⊢ (𝜑 → 𝜃) |
12 | keephyp2v.6 | . . 3 ⊢ 𝜏 | |
13 | iffalse 4045 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶) | |
14 | 13 | eqcomd 2616 | . . . . 5 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐴, 𝐶)) |
15 | keephyp2v.3 | . . . . 5 ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜂)) |
17 | iffalse 4045 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐷) | |
18 | 17 | eqcomd 2616 | . . . . 5 ⊢ (¬ 𝜑 → 𝐷 = if(𝜑, 𝐵, 𝐷)) |
19 | keephyp2v.4 | . . . . 5 ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜃)) |
21 | 16, 20 | bitrd 267 | . . 3 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜃)) |
22 | 12, 21 | mpbii 222 | . 2 ⊢ (¬ 𝜑 → 𝜃) |
23 | 11, 22 | pm2.61i 175 | 1 ⊢ 𝜃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = 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: (None) |
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