Theorem keephyp3v 4104

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 4042	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐴)
3	2	eqcomd 2616	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐷))
4		keephyp3v.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌 ↔ 𝜒))
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝜌 ↔ 𝜒))
6		iftrue 4042	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑅) = 𝐵)
7	6	eqcomd 2616	. . . . 5 ⊢ (𝜑 → 𝐵 = if(𝜑, 𝐵, 𝑅))
8		keephyp3v.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
10		iftrue 4042	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑆) = 𝐶)
11	10	eqcomd 2616	. . . . 5 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐶, 𝑆))
12		keephyp3v.3	. . . . 5 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏))
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
14	5, 9, 13	3bitrd 293	. . 3 ⊢ (𝜑 → (𝜌 ↔ 𝜏))
15	1, 14	mpbii 222	. 2 ⊢ (𝜑 → 𝜏)
16		keephyp3v.8	. . 3 ⊢ 𝜂
17		iffalse 4045	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐷)
18	17	eqcomd 2616	. . . . 5 ⊢ (¬ 𝜑 → 𝐷 = if(𝜑, 𝐴, 𝐷))
19		keephyp3v.4	. . . . 5 ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁))
20	18, 19	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜁))
21		iffalse 4045	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑅) = 𝑅)
22	21	eqcomd 2616	. . . . 5 ⊢ (¬ 𝜑 → 𝑅 = if(𝜑, 𝐵, 𝑅))
23		keephyp3v.5	. . . . 5 ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎))
24	22, 23	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜁 ↔ 𝜎))
25		iffalse 4045	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑆) = 𝑆)
26	25	eqcomd 2616	. . . . 5 ⊢ (¬ 𝜑 → 𝑆 = if(𝜑, 𝐶, 𝑆))
27		keephyp3v.6	. . . . 5 ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏))
28	26, 27	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜎 ↔ 𝜏))
29	20, 24, 28	3bitrd 293	. . 3 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜏))
30	16, 29	mpbii 222	. 2 ⊢ (¬ 𝜑 → 𝜏)
31	15, 30	pm2.61i 175	1 ⊢ 𝜏

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037

This theorem is referenced by:  sseliALT  4719