Theorem elimhyp4v 3714

Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3704). (Contributed by NM, 16-Apr-2005.)

Hypotheses

Ref	Expression
elimhyp4v.1	⊢ (A = if(φ, A, D) → (φ ↔ χ))
elimhyp4v.2	⊢ (B = if(φ, B, R) → (χ ↔ θ))
elimhyp4v.3	⊢ (C = if(φ, C, S) → (θ ↔ τ))
elimhyp4v.4	⊢ (F = if(φ, F, G) → (τ ↔ ψ))
elimhyp4v.5	⊢ (D = if(φ, A, D) → (η ↔ ζ))
elimhyp4v.6	⊢ (R = if(φ, B, R) → (ζ ↔ σ))
elimhyp4v.7	⊢ (S = if(φ, C, S) → (σ ↔ ρ))
elimhyp4v.8	⊢ (G = if(φ, F, G) → (ρ ↔ ψ))
elimhyp4v.9	⊢ η

Assertion

Ref	Expression
elimhyp4v	⊢ ψ

Proof of Theorem elimhyp4v

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . . . . 7 ⊢ (φ → if(φ, A, D) = A)
2	1	eqcomd 2358	. . . . . 6 ⊢ (φ → A = if(φ, A, D))
3		elimhyp4v.1	. . . . . 6 ⊢ (A = if(φ, A, D) → (φ ↔ χ))
4	2, 3	syl 15	. . . . 5 ⊢ (φ → (φ ↔ χ))
5		iftrue 3669	. . . . . . 7 ⊢ (φ → if(φ, B, R) = B)
6	5	eqcomd 2358	. . . . . 6 ⊢ (φ → B = if(φ, B, R))
7		elimhyp4v.2	. . . . . 6 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
8	6, 7	syl 15	. . . . 5 ⊢ (φ → (χ ↔ θ))
9	4, 8	bitrd 244	. . . 4 ⊢ (φ → (φ ↔ θ))
10		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
11	10	eqcomd 2358	. . . . 5 ⊢ (φ → C = if(φ, C, S))
12		elimhyp4v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
13	11, 12	syl 15	. . . 4 ⊢ (φ → (θ ↔ τ))
14		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, F, G) = F)
15	14	eqcomd 2358	. . . . 5 ⊢ (φ → F = if(φ, F, G))
16		elimhyp4v.4	. . . . 5 ⊢ (F = if(φ, F, G) → (τ ↔ ψ))
17	15, 16	syl 15	. . . 4 ⊢ (φ → (τ ↔ ψ))
18	9, 13, 17	3bitrd 270	. . 3 ⊢ (φ → (φ ↔ ψ))
19	18	ibi 232	. 2 ⊢ (φ → ψ)
20		elimhyp4v.9	. . 3 ⊢ η
21		iffalse 3670	. . . . . . 7 ⊢ (¬ φ → if(φ, A, D) = D)
22	21	eqcomd 2358	. . . . . 6 ⊢ (¬ φ → D = if(φ, A, D))
23		elimhyp4v.5	. . . . . 6 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
24	22, 23	syl 15	. . . . 5 ⊢ (¬ φ → (η ↔ ζ))
25		iffalse 3670	. . . . . . 7 ⊢ (¬ φ → if(φ, B, R) = R)
26	25	eqcomd 2358	. . . . . 6 ⊢ (¬ φ → R = if(φ, B, R))
27		elimhyp4v.6	. . . . . 6 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
28	26, 27	syl 15	. . . . 5 ⊢ (¬ φ → (ζ ↔ σ))
29	24, 28	bitrd 244	. . . 4 ⊢ (¬ φ → (η ↔ σ))
30		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
31	30	eqcomd 2358	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
32		elimhyp4v.7	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ ρ))
33	31, 32	syl 15	. . . 4 ⊢ (¬ φ → (σ ↔ ρ))
34		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, F, G) = G)
35	34	eqcomd 2358	. . . . 5 ⊢ (¬ φ → G = if(φ, F, G))
36		elimhyp4v.8	. . . . 5 ⊢ (G = if(φ, F, G) → (ρ ↔ ψ))
37	35, 36	syl 15	. . . 4 ⊢ (¬ φ → (ρ ↔ ψ))
38	29, 33, 37	3bitrd 270	. . 3 ⊢ (¬ φ → (η ↔ ψ))
39	20, 38	mpbii 202	. 2 ⊢ (¬ φ → ψ)
40	19, 39	pm2.61i 156	1 ⊢ ψ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = 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: (None)