Theorem dedth2v 3707

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.)

Hypotheses

Ref	Expression
dedth2v.1	⊢ (A = if(φ, A, C) → (ψ ↔ χ))
dedth2v.2	⊢ (B = if(φ, B, D) → (χ ↔ θ))
dedth2v.3	⊢ θ

Assertion

Ref	Expression
dedth2v	⊢ (φ → ψ)

Proof of Theorem dedth2v

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)

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)