Theorem elimhyp2v

Description: Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999)

	Ref	Expression
Hypotheses	elimhyp2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜑 ↔ 𝜒 ) )
	elimhyp2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
	elimhyp2v.3	⊢ ( 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜏 ↔ 𝜂 ) )
	elimhyp2v.4	⊢ ( 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜂 ↔ 𝜃 ) )
	elimhyp2v.5	⊢ 𝜏
Assertion	elimhyp2v	⊢ 𝜃

Proof

Step	Hyp	Ref	Expression
1		elimhyp2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜑 ↔ 𝜒 ) )
2		elimhyp2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
3		elimhyp2v.3	⊢ ( 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜏 ↔ 𝜂 ) )
4		elimhyp2v.4	⊢ ( 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜂 ↔ 𝜃 ) )
5		elimhyp2v.5	⊢ 𝜏
6		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐶 ) = 𝐴 )
7	6	eqcomd	⊢ ( 𝜑 → 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) )
8	7 1	syl	⊢ ( 𝜑 → ( 𝜑 ↔ 𝜒 ) )
9		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐵 , 𝐷 ) = 𝐵 )
10	9	eqcomd	⊢ ( 𝜑 → 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) )
11	10 2	syl	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
12	8 11	bitrd	⊢ ( 𝜑 → ( 𝜑 ↔ 𝜃 ) )
13	12	ibi	⊢ ( 𝜑 → 𝜃 )
14		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐶 ) = 𝐶 )
15	14	eqcomd	⊢ ( ¬ 𝜑 → 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) )
16	15 3	syl	⊢ ( ¬ 𝜑 → ( 𝜏 ↔ 𝜂 ) )
17		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐵 , 𝐷 ) = 𝐷 )
18	17	eqcomd	⊢ ( ¬ 𝜑 → 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) )
19	18 4	syl	⊢ ( ¬ 𝜑 → ( 𝜂 ↔ 𝜃 ) )
20	16 19	bitrd	⊢ ( ¬ 𝜑 → ( 𝜏 ↔ 𝜃 ) )
21	5 20	mpbii	⊢ ( ¬ 𝜑 → 𝜃 )
22	13 21	pm2.61i	⊢ 𝜃