Theorem elimhyp2v 1791

Description: Eliminate a hypothesis containing 2 class variables.

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 1780	. . . . . 6 |- (ph -> if(ph, A, C) = A)
2	1	cleqcomd 1106	. . . . 5 |- (ph -> A = if(ph, A, C))
3		elimhyp2v.1	. . . . 5 |- (A = if(ph, A, C) -> (ph <-> ch))
4	2, 3	syl 12	. . . 4 |- (ph -> (ph <-> ch))
5		iftrue 1780	. . . . . 6 |- (ph -> if(ph, B, D) = B)
6	5	cleqcomd 1106	. . . . 5 |- (ph -> B = if(ph, B, D))
7		elimhyp2v.2	. . . . 5 |- (B = if(ph, B, D) -> (ch <-> th))
8	6, 7	syl 12	. . . 4 |- (ph -> (ch <-> th))
9	4, 8	bitrd 406	. . 3 |- (ph -> (ph <-> th))
10	9	ibi 449	. 2 |- (ph -> th)
11		elimhyp2v.5	. . 3 |- ta
12		iffalse 1781	. . . . . 6 |- (-. ph -> if(ph, A, C) = C)
13	12	cleqcomd 1106	. . . . 5 |- (-. ph -> C = if(ph, A, C))
14		elimhyp2v.3	. . . . 5 |- (C = if(ph, A, C) -> (ta <-> et))
15	13, 14	syl 12	. . . 4 |- (-. ph -> (ta <-> et))
16		iffalse 1781	. . . . . 6 |- (-. ph -> if(ph, B, D) = D)
17	16	cleqcomd 1106	. . . . 5 |- (-. ph -> D = if(ph, B, D))
18		elimhyp2v.4	. . . . 5 |- (D = if(ph, B, D) -> (et <-> th))
19	17, 18	syl 12	. . . 4 |- (-. ph -> (et <-> th))
20	15, 19	bitrd 406	. . 3 |- (-. ph -> (ta <-> th))
21	11, 20	mpbii 168	. 2 |- (-. ph -> th)
22	10, 21	pm2.61i 110	1 |- th

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   = wceq 1091  ifcif 1776

This theorem is referenced by:  hlimcau 5142  omls 5251  osumlem8 5537

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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