Theorem elimhyp3v

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

	Ref	Expression
Hypotheses	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	elimhyp3v	⊢ 𝜏

Proof

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