Theorem dedth4h 3707

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3705. (Contributed by NM, 16-May-1999.)

Hypotheses

Ref	Expression
dedth4h.1	⊢ (A = if(φ, A, R) → (τ ↔ η))
dedth4h.2	⊢ (B = if(ψ, B, S) → (η ↔ ζ))
dedth4h.3	⊢ (C = if(χ, C, F) → (ζ ↔ σ))
dedth4h.4	⊢ (D = if(θ, D, G) → (σ ↔ ρ))
dedth4h.5	⊢ ρ

Assertion

Ref	Expression
dedth4h	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)

Proof of Theorem dedth4h

Step	Hyp	Ref	Expression
1		dedth4h.1	. . . 4 ⊢ (A = if(φ, A, R) → (τ ↔ η))
2	1	imbi2d 307	. . 3 ⊢ (A = if(φ, A, R) → (((χ ∧ θ) → τ) ↔ ((χ ∧ θ) → η)))
3		dedth4h.2	. . . 4 ⊢ (B = if(ψ, B, S) → (η ↔ ζ))
4	3	imbi2d 307	. . 3 ⊢ (B = if(ψ, B, S) → (((χ ∧ θ) → η) ↔ ((χ ∧ θ) → ζ)))
5		dedth4h.3	. . . 4 ⊢ (C = if(χ, C, F) → (ζ ↔ σ))
6		dedth4h.4	. . . 4 ⊢ (D = if(θ, D, G) → (σ ↔ ρ))
7		dedth4h.5	. . . 4 ⊢ ρ
8	5, 6, 7	dedth2h 3705	. . 3 ⊢ ((χ ∧ θ) → ζ)
9	2, 4, 8	dedth2h 3705	. 2 ⊢ ((φ ∧ ψ) → ((χ ∧ θ) → τ))
10	9	imp 418	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = 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-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:  dedth4v  3710