Theorem elimhyp2v 3712

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

Hypotheses

Ref	Expression
elimhyp2v.1	|- (A = if(ph, A, C) -> (ph <-> ch))
elimhyp2v.2	|- (B = if(ph, B, D) -> (ch <-> th))
elimhyp2v.3	|- (C = if(ph, A, C) -> (ta <-> et))
elimhyp2v.4	|- (D = if(ph, B, D) -> (et <-> th))
elimhyp2v.5	|- ta

Assertion

Ref	Expression
elimhyp2v	|- th

Proof of Theorem elimhyp2v

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . . . 6 |- (ph -> if(ph, A, C) = A)
2	1	eqcomd 2358	. . . . 5 |- (ph -> A = if(ph, A, C))
3		elimhyp2v.1	. . . . 5 |- (A = if(ph, A, C) -> (ph <-> ch))
4	2, 3	syl 15	. . . 4 |- (ph -> (ph <-> ch))
5		iftrue 3669	. . . . . 6 |- (ph -> if(ph, B, D) = B)
6	5	eqcomd 2358	. . . . 5 |- (ph -> B = if(ph, B, D))
7		elimhyp2v.2	. . . . 5 |- (B = if(ph, B, D) -> (ch <-> th))
8	6, 7	syl 15	. . . 4 |- (ph -> (ch <-> th))
9	4, 8	bitrd 244	. . 3 |- (ph -> (ph <-> th))
10	9	ibi 232	. 2 |- (ph -> th)
11		elimhyp2v.5	. . 3 |- ta
12		iffalse 3670	. . . . . 6 |- (-. ph -> if(ph, A, C) = C)
13	12	eqcomd 2358	. . . . 5 |- (-. ph -> C = if(ph, A, C))
14		elimhyp2v.3	. . . . 5 |- (C = if(ph, A, C) -> (ta <-> et))
15	13, 14	syl 15	. . . 4 |- (-. ph -> (ta <-> et))
16		iffalse 3670	. . . . . 6 |- (-. ph -> if(ph, B, D) = D)
17	16	eqcomd 2358	. . . . 5 |- (-. ph -> D = if(ph, B, D))
18		elimhyp2v.4	. . . . 5 |- (D = if(ph, B, D) -> (et <-> th))
19	17, 18	syl 15	. . . 4 |- (-. ph -> (et <-> th))
20	15, 19	bitrd 244	. . 3 |- (-. ph -> (ta <-> th))
21	11, 20	mpbii 202	. 2 |- (-. ph -> th)
22	10, 21	pm2.61i 156	1 |- th

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642  ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by: (None)