Theorem dedth4h 4567

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4565. (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 340	. . 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 340	. . 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 4565	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜁)
9	2, 4, 8	dedth2h 4565	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏))
10	9	imp 406	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ifcif 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-if 4506

This theorem is referenced by:  dedth4v  4570  fprg  7150  omopth  8679  nn0opth2  14295  ax5seglem8  28920  hvsubsub4  31046  norm3lemt  31138  eigorth  31824