Theorem dedth2h 3705

Description: Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3708 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3704. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
dedth2h.1	⊢ (A = if(φ, A, C) → (χ ↔ θ))
dedth2h.2	⊢ (B = if(ψ, B, D) → (θ ↔ τ))
dedth2h.3	⊢ τ

Assertion

Ref	Expression
dedth2h	⊢ ((φ ∧ ψ) → χ)

Proof of Theorem dedth2h

Step	Hyp	Ref	Expression
1		dedth2h.1	. . . 4 ⊢ (A = if(φ, A, C) → (χ ↔ θ))
2	1	imbi2d 307	. . 3 ⊢ (A = if(φ, A, C) → ((ψ → χ) ↔ (ψ → θ)))
3		dedth2h.2	. . . 4 ⊢ (B = if(ψ, B, D) → (θ ↔ τ))
4		dedth2h.3	. . . 4 ⊢ τ
5	3, 4	dedth 3704	. . 3 ⊢ (ψ → θ)
6	2, 5	dedth 3704	. 2 ⊢ (φ → (ψ → χ))
7	6	imp 418	1 ⊢ ((φ ∧ ψ) → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ifcif 3663

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 3664

This theorem is referenced by:  dedth3h  3706  dedth4h  3707  dedth2v  3708