Theorem keephyp3v 2388

Description: Keep a hypothesis containing 3 class variables.

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 2356	. . . . . 6 ⊢ (φ → if(φ, A, D) = A)
3	2	eqcomd 1472	. . . . 5 ⊢ (φ → A = if(φ, A, D))
4		keephyp3v.1	. . . . 5 ⊢ (A = if(φ, A, D) → (ρ ↔ χ))
5	3, 4	syl 10	. . . 4 ⊢ (φ → (ρ ↔ χ))
6		iftrue 2356	. . . . . 6 ⊢ (φ → if(φ, B, R) = B)
7	6	eqcomd 1472	. . . . 5 ⊢ (φ → B = if(φ, B, R))
8		keephyp3v.2	. . . . 5 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
9	7, 8	syl 10	. . . 4 ⊢ (φ → (χ ↔ θ))
10		iftrue 2356	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
11	10	eqcomd 1472	. . . . 5 ⊢ (φ → C = if(φ, C, S))
12		keephyp3v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
13	11, 12	syl 10	. . . 4 ⊢ (φ → (θ ↔ τ))
14	5, 9, 13	3bitrd 542	. . 3 ⊢ (φ → (ρ ↔ τ))
15	1, 14	mpbii 193	. 2 ⊢ (φ → τ)
16		keephyp3v.8	. . 3 ⊢ η
17		iffalse 2357	. . . . . 6 ⊢ (¬ φ → if(φ, A, D) = D)
18	17	eqcomd 1472	. . . . 5 ⊢ (¬ φ → D = if(φ, A, D))
19		keephyp3v.4	. . . . 5 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
20	18, 19	syl 10	. . . 4 ⊢ (¬ φ → (η ↔ ζ))
21		iffalse 2357	. . . . . 6 ⊢ (¬ φ → if(φ, B, R) = R)
22	21	eqcomd 1472	. . . . 5 ⊢ (¬ φ → R = if(φ, B, R))
23		keephyp3v.5	. . . . 5 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
24	22, 23	syl 10	. . . 4 ⊢ (¬ φ → (ζ ↔ σ))
25		iffalse 2357	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
26	25	eqcomd 1472	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
27		keephyp3v.6	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ τ))
28	26, 27	syl 10	. . . 4 ⊢ (¬ φ → (σ ↔ τ))
29	20, 24, 28	3bitrd 542	. . 3 ⊢ (¬ φ → (η ↔ τ))
30	16, 29	mpbii 193	. 2 ⊢ (¬ φ → τ)
31	15, 30	pm2.61i 126	1 ⊢ τ

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   = wceq 953   ifcif 2351

This theorem is referenced by:  projlem7 9108

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

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