Theorem elimhyp2v 4147

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

Hypotheses

Ref	Expression
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

Ref	Expression
elimhyp2v	⊢ 𝜃

Proof of Theorem elimhyp2v

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐴)
2	1	eqcomd 2628	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐶))
3		elimhyp2v.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑 ↔ 𝜒))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝜒))
5		iftrue 4092	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐵)
6	5	eqcomd 2628	. . . . 5 ⊢ (𝜑 → 𝐵 = if(𝜑, 𝐵, 𝐷))
7		elimhyp2v.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
9	4, 8	bitrd 268	. . 3 ⊢ (𝜑 → (𝜑 ↔ 𝜃))
10	9	ibi 256	. 2 ⊢ (𝜑 → 𝜃)
11		elimhyp2v.5	. . 3 ⊢ 𝜏
12		iffalse 4095	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶)
13	12	eqcomd 2628	. . . . 5 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐴, 𝐶))
14		elimhyp2v.3	. . . . 5 ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂))
15	13, 14	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜂))
16		iffalse 4095	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐷)
17	16	eqcomd 2628	. . . . 5 ⊢ (¬ 𝜑 → 𝐷 = if(𝜑, 𝐵, 𝐷))
18		elimhyp2v.4	. . . . 5 ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃))
19	17, 18	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜃))
20	15, 19	bitrd 268	. . 3 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜃))
21	11, 20	mpbii 223	. 2 ⊢ (¬ 𝜑 → 𝜃)
22	10, 21	pm2.61i 176	1 ⊢ 𝜃

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  omlsi  28263