Theorem elimhyp3v 4550

Description: Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

Hypotheses

Ref	Expression
elimhyp3v.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒))
elimhyp3v.2	⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃))
elimhyp3v.3	⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏))
elimhyp3v.4	⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁))
elimhyp3v.5	⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎))
elimhyp3v.6	⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏))
elimhyp3v.7	⊢ 𝜂

Assertion

Ref	Expression
elimhyp3v	⊢ 𝜏

Proof of Theorem elimhyp3v

Step	Hyp	Ref	Expression
1		iftrue 4488	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐴)
2	1	eqcomd 2767	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐷))
3		elimhyp3v.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒))
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝜒))
5		iftrue 4488	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑅) = 𝐵)
6	5	eqcomd 2767	. . . . 5 ⊢ (𝜑 → 𝐵 = if(𝜑, 𝐵, 𝑅))
7		elimhyp3v.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃))
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
9		iftrue 4488	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑆) = 𝐶)
10	9	eqcomd 2767	. . . . 5 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐶, 𝑆))
11		elimhyp3v.3	. . . . 5 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏))
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
13	4, 8, 12	3bitrd 308	. . 3 ⊢ (𝜑 → (𝜑 ↔ 𝜏))
14	13	ibi 270	. 2 ⊢ (𝜑 → 𝜏)
15		elimhyp3v.7	. . 3 ⊢ 𝜂
16		iffalse 4491	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐷)
17	16	eqcomd 2767	. . . . 5 ⊢ (¬ 𝜑 → 𝐷 = if(𝜑, 𝐴, 𝐷))
18		elimhyp3v.4	. . . . 5 ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁))
19	17, 18	syl 18	. . . 4 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜁))
20		iffalse 4491	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑅) = 𝑅)
21	20	eqcomd 2767	. . . . 5 ⊢ (¬ 𝜑 → 𝑅 = if(𝜑, 𝐵, 𝑅))
22		elimhyp3v.5	. . . . 5 ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎))
23	21, 22	syl 18	. . . 4 ⊢ (¬ 𝜑 → (𝜁 ↔ 𝜎))
24		iffalse 4491	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑆) = 𝑆)
25	24	eqcomd 2767	. . . . 5 ⊢ (¬ 𝜑 → 𝑆 = if(𝜑, 𝐶, 𝑆))
26		elimhyp3v.6	. . . . 5 ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏))
27	25, 26	syl 18	. . . 4 ⊢ (¬ 𝜑 → (𝜎 ↔ 𝜏))
28	19, 23, 27	3bitrd 308	. . 3 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜏))
29	15, 28	mpbii 236	. 2 ⊢ (¬ 𝜑 → 𝜏)
30	14, 29	pm2.61i 184	1 ⊢ 𝜏

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4483

This theorem is used by:  sseliALT  5263