New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dedth2v | GIF version |
Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3704 is simpler to use. See also comments in dedth 3703. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
Ref | Expression |
---|---|
dedth2v.1 | ⊢ (A = if(φ, A, C) → (ψ ↔ χ)) |
dedth2v.2 | ⊢ (B = if(φ, B, D) → (χ ↔ θ)) |
dedth2v.3 | ⊢ θ |
Ref | Expression |
---|---|
dedth2v | ⊢ (φ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth2v.1 | . . 3 ⊢ (A = if(φ, A, C) → (ψ ↔ χ)) | |
2 | dedth2v.2 | . . 3 ⊢ (B = if(φ, B, D) → (χ ↔ θ)) | |
3 | dedth2v.3 | . . 3 ⊢ θ | |
4 | 1, 2, 3 | dedth2h 3704 | . 2 ⊢ ((φ ∧ φ) → ψ) |
5 | 4 | anidms 626 | 1 ⊢ (φ → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → 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: (None) |
Copyright terms: Public domain | W3C validator |