Theorem dedth3v 3709

Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3708. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth3v.1	⊢ (A = if(φ, A, D) → (ψ ↔ χ))
dedth3v.2	⊢ (B = if(φ, B, R) → (χ ↔ θ))
dedth3v.3	⊢ (C = if(φ, C, S) → (θ ↔ τ))
dedth3v.4	⊢ τ

Assertion

Ref	Expression
dedth3v	⊢ (φ → ψ)

Proof of Theorem dedth3v

Step	Hyp	Ref	Expression
1		dedth3v.1	. . . 4 ⊢ (A = if(φ, A, D) → (ψ ↔ χ))
2		dedth3v.2	. . . 4 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
3		dedth3v.3	. . . 4 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
4		dedth3v.4	. . . 4 ⊢ τ
5	1, 2, 3, 4	dedth3h 3706	. . 3 ⊢ ((φ ∧ φ ∧ φ) → ψ)
6	5	3anidm12 1239	. 2 ⊢ ((φ ∧ φ) → ψ)
7	6	anidms 626	1 ⊢ (φ → ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3663

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is used by: (None)