Theorem keephyp2v 4531

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4517). (Contributed by NM, 16-Apr-2005.)

Hypotheses

Ref	Expression
keephyp2v.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒))
keephyp2v.2	⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))
keephyp2v.3	⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂))
keephyp2v.4	⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃))
keephyp2v.5	⊢ 𝜓
keephyp2v.6	⊢ 𝜏

Assertion

Ref	Expression
keephyp2v	⊢ 𝜃

Proof of Theorem keephyp2v

Step	Hyp	Ref	Expression
1		keephyp2v.5	. . 3 ⊢ 𝜓
2		iftrue 4465	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐴)
3	2	eqcomd 2744	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐶))
4		keephyp2v.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒))
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
6		iftrue 4465	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐵)
7	6	eqcomd 2744	. . . . 5 ⊢ (𝜑 → 𝐵 = if(𝜑, 𝐵, 𝐷))
8		keephyp2v.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
10	5, 9	bitrd 278	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
11	1, 10	mpbii 232	. 2 ⊢ (𝜑 → 𝜃)
12		keephyp2v.6	. . 3 ⊢ 𝜏
13		iffalse 4468	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶)
14	13	eqcomd 2744	. . . . 5 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐴, 𝐶))
15		keephyp2v.3	. . . . 5 ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂))
16	14, 15	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜂))
17		iffalse 4468	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐷)
18	17	eqcomd 2744	. . . . 5 ⊢ (¬ 𝜑 → 𝐷 = if(𝜑, 𝐵, 𝐷))
19		keephyp2v.4	. . . . 5 ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃))
20	18, 19	syl 17	. . . 4 ⊢ (¬ 𝜑 → (𝜂 ↔ 𝜃))
21	16, 20	bitrd 278	. . 3 ⊢ (¬ 𝜑 → (𝜏 ↔ 𝜃))
22	12, 21	mpbii 232	. 2 ⊢ (¬ 𝜑 → 𝜃)
23	11, 22	pm2.61i 182	1 ⊢ 𝜃

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-if 4460

This theorem is referenced by: (None)