Theorem dedth3h 3706

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3705. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
dedth3h.1	⊢ (A = if(φ, A, D) → (θ ↔ τ))
dedth3h.2	⊢ (B = if(ψ, B, R) → (τ ↔ η))
dedth3h.3	⊢ (C = if(χ, C, S) → (η ↔ ζ))
dedth3h.4	⊢ ζ

Assertion

Ref	Expression
dedth3h	⊢ ((φ ∧ ψ ∧ χ) → θ)

Proof of Theorem dedth3h

Step	Hyp	Ref	Expression
1		dedth3h.1	. . . 4 ⊢ (A = if(φ, A, D) → (θ ↔ τ))
2	1	imbi2d 307	. . 3 ⊢ (A = if(φ, A, D) → (((ψ ∧ χ) → θ) ↔ ((ψ ∧ χ) → τ)))
3		dedth3h.2	. . . 4 ⊢ (B = if(ψ, B, R) → (τ ↔ η))
4		dedth3h.3	. . . 4 ⊢ (C = if(χ, C, S) → (η ↔ ζ))
5		dedth3h.4	. . . 4 ⊢ ζ
6	3, 4, 5	dedth2h 3705	. . 3 ⊢ ((ψ ∧ χ) → τ)
7	2, 6	dedth 3704	. 2 ⊢ (φ → ((ψ ∧ χ) → θ))
8	7	3impib 1149	1 ⊢ ((φ ∧ ψ ∧ χ) → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = 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-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 referenced by:  dedth3v  3709