Theorem keephyp3v 1795

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 1780	. . . . . 6 ⊢ (φ → if(φ, A, D) = A)
3	2	cleqcomd 1106	. . . . 5 ⊢ (φ → A = if(φ, A, D))
4		keephyp3v.1	. . . . 5 ⊢ (A = if(φ, A, D) → (ρ ↔ χ))
5	3, 4	syl 12	. . . 4 ⊢ (φ → (ρ ↔ χ))
6		iftrue 1780	. . . . . 6 ⊢ (φ → if(φ, B, R) = B)
7	6	cleqcomd 1106	. . . . 5 ⊢ (φ → B = if(φ, B, R))
8		keephyp3v.2	. . . . 5 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
9	7, 8	syl 12	. . . 4 ⊢ (φ → (χ ↔ θ))
10		iftrue 1780	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
11	10	cleqcomd 1106	. . . . 5 ⊢ (φ → C = if(φ, C, S))
12		keephyp3v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
13	11, 12	syl 12	. . . 4 ⊢ (φ → (θ ↔ τ))
14	5, 9, 13	3bitrd 422	. . 3 ⊢ (φ → (ρ ↔ τ))
15	1, 14	mpbii 168	. 2 ⊢ (φ → τ)
16		keephyp3v.8	. . 3 ⊢ η
17		iffalse 1781	. . . . . 6 ⊢ (¬ φ → if(φ, A, D) = D)
18	17	cleqcomd 1106	. . . . 5 ⊢ (¬ φ → D = if(φ, A, D))
19		keephyp3v.4	. . . . 5 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
20	18, 19	syl 12	. . . 4 ⊢ (¬ φ → (η ↔ ζ))
21		iffalse 1781	. . . . . 6 ⊢ (¬ φ → if(φ, B, R) = R)
22	21	cleqcomd 1106	. . . . 5 ⊢ (¬ φ → R = if(φ, B, R))
23		keephyp3v.5	. . . . 5 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
24	22, 23	syl 12	. . . 4 ⊢ (¬ φ → (ζ ↔ σ))
25		iffalse 1781	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
26	25	cleqcomd 1106	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
27		keephyp3v.6	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ τ))
28	26, 27	syl 12	. . . 4 ⊢ (¬ φ → (σ ↔ τ))
29	20, 24, 28	3bitrd 422	. . 3 ⊢ (¬ φ → (η ↔ τ))
30	16, 29	mpbii 168	. 2 ⊢ (¬ φ → τ)
31	15, 30	pm2.61i 110	1 ⊢ τ

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091  ifcif 1776

This theorem is referenced by:  projlem7 5199

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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