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Mirrors > Home > NFE Home > Th. List > elimhyp | GIF version |
Description: Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3703. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
elimhyp.1 | ⊢ (A = if(φ, A, B) → (φ ↔ ψ)) |
elimhyp.2 | ⊢ (B = if(φ, A, B) → (χ ↔ ψ)) |
elimhyp.3 | ⊢ χ |
Ref | Expression |
---|---|
elimhyp | ⊢ ψ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3668 | . . . . 5 ⊢ (φ → if(φ, A, B) = A) | |
2 | 1 | eqcomd 2358 | . . . 4 ⊢ (φ → A = if(φ, A, B)) |
3 | elimhyp.1 | . . . 4 ⊢ (A = if(φ, A, B) → (φ ↔ ψ)) | |
4 | 2, 3 | syl 15 | . . 3 ⊢ (φ → (φ ↔ ψ)) |
5 | 4 | ibi 232 | . 2 ⊢ (φ → ψ) |
6 | elimhyp.3 | . . 3 ⊢ χ | |
7 | iffalse 3669 | . . . . 5 ⊢ (¬ φ → if(φ, A, B) = B) | |
8 | 7 | eqcomd 2358 | . . . 4 ⊢ (¬ φ → B = if(φ, A, B)) |
9 | elimhyp.2 | . . . 4 ⊢ (B = if(φ, A, B) → (χ ↔ ψ)) | |
10 | 8, 9 | syl 15 | . . 3 ⊢ (¬ φ → (χ ↔ ψ)) |
11 | 6, 10 | mpbii 202 | . 2 ⊢ (¬ φ → ψ) |
12 | 5, 11 | pm2.61i 156 | 1 ⊢ ψ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 = wceq 1642 ifcif 3662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3663 |
This theorem is referenced by: elimel 3714 elimf 5222 |
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