Theorem keephyp3v 3719

Description: Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)

Hypotheses

Ref	Expression
keephyp3v.1	⊢ (A = if(φ, A, D) → (ρ ↔ χ))
keephyp3v.2	⊢ (B = if(φ, B, R) → (χ ↔ θ))
keephyp3v.3	⊢ (C = if(φ, C, S) → (θ ↔ τ))
keephyp3v.4	⊢ (D = if(φ, A, D) → (η ↔ ζ))
keephyp3v.5	⊢ (R = if(φ, B, R) → (ζ ↔ σ))
keephyp3v.6	⊢ (S = if(φ, C, S) → (σ ↔ τ))
keephyp3v.7	⊢ ρ
keephyp3v.8	⊢ η

Assertion

Ref	Expression
keephyp3v	⊢ τ

Proof of Theorem keephyp3v

Step	Hyp	Ref	Expression
1		keephyp3v.7	. . 3 ⊢ ρ
2		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, A, D) = A)
3	2	eqcomd 2358	. . . . 5 ⊢ (φ → A = if(φ, A, D))
4		keephyp3v.1	. . . . 5 ⊢ (A = if(φ, A, D) → (ρ ↔ χ))
5	3, 4	syl 15	. . . 4 ⊢ (φ → (ρ ↔ χ))
6		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, B, R) = B)
7	6	eqcomd 2358	. . . . 5 ⊢ (φ → B = if(φ, B, R))
8		keephyp3v.2	. . . . 5 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
9	7, 8	syl 15	. . . 4 ⊢ (φ → (χ ↔ θ))
10		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
11	10	eqcomd 2358	. . . . 5 ⊢ (φ → C = if(φ, C, S))
12		keephyp3v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
13	11, 12	syl 15	. . . 4 ⊢ (φ → (θ ↔ τ))
14	5, 9, 13	3bitrd 270	. . 3 ⊢ (φ → (ρ ↔ τ))
15	1, 14	mpbii 202	. 2 ⊢ (φ → τ)
16		keephyp3v.8	. . 3 ⊢ η
17		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, A, D) = D)
18	17	eqcomd 2358	. . . . 5 ⊢ (¬ φ → D = if(φ, A, D))
19		keephyp3v.4	. . . . 5 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
20	18, 19	syl 15	. . . 4 ⊢ (¬ φ → (η ↔ ζ))
21		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, B, R) = R)
22	21	eqcomd 2358	. . . . 5 ⊢ (¬ φ → R = if(φ, B, R))
23		keephyp3v.5	. . . . 5 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
24	22, 23	syl 15	. . . 4 ⊢ (¬ φ → (ζ ↔ σ))
25		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
26	25	eqcomd 2358	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
27		keephyp3v.6	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ τ))
28	26, 27	syl 15	. . . 4 ⊢ (¬ φ → (σ ↔ τ))
29	20, 24, 28	3bitrd 270	. . 3 ⊢ (¬ φ → (η ↔ τ))
30	16, 29	mpbii 202	. 2 ⊢ (¬ φ → τ)
31	15, 30	pm2.61i 156	1 ⊢ τ

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)