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Mirrors > Home > NFE Home > Th. List > dedth4v | GIF version |
Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3707. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
Ref | Expression |
---|---|
dedth4v.1 | ⊢ (A = if(φ, A, R) → (ψ ↔ χ)) |
dedth4v.2 | ⊢ (B = if(φ, B, S) → (χ ↔ θ)) |
dedth4v.3 | ⊢ (C = if(φ, C, T) → (θ ↔ τ)) |
dedth4v.4 | ⊢ (D = if(φ, D, U) → (τ ↔ η)) |
dedth4v.5 | ⊢ η |
Ref | Expression |
---|---|
dedth4v | ⊢ (φ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth4v.1 | . . . 4 ⊢ (A = if(φ, A, R) → (ψ ↔ χ)) | |
2 | dedth4v.2 | . . . 4 ⊢ (B = if(φ, B, S) → (χ ↔ θ)) | |
3 | dedth4v.3 | . . . 4 ⊢ (C = if(φ, C, T) → (θ ↔ τ)) | |
4 | dedth4v.4 | . . . 4 ⊢ (D = if(φ, D, U) → (τ ↔ η)) | |
5 | dedth4v.5 | . . . 4 ⊢ η | |
6 | 1, 2, 3, 4, 5 | dedth4h 3706 | . . 3 ⊢ (((φ ∧ φ) ∧ (φ ∧ φ)) → ψ) |
7 | 6 | anidms 626 | . 2 ⊢ ((φ ∧ φ) → ψ) |
8 | 7 | anidms 626 | 1 ⊢ (φ → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = 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) |
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