Theorem elimhyp2v 2381

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 2356	. . . . . 6 |- (ph -> if(ph, A, C) = A)
2	1	eqcomd 1472	. . . . 5 |- (ph -> A = if(ph, A, C))
3		elimhyp2v.1	. . . . 5 |- (A = if(ph, A, C) -> (ph <-> ch))
4	2, 3	syl 10	. . . 4 |- (ph -> (ph <-> ch))
5		iftrue 2356	. . . . . 6 |- (ph -> if(ph, B, D) = B)
6	5	eqcomd 1472	. . . . 5 |- (ph -> B = if(ph, B, D))
7		elimhyp2v.2	. . . . 5 |- (B = if(ph, B, D) -> (ch <-> th))
8	6, 7	syl 10	. . . 4 |- (ph -> (ch <-> th))
9	4, 8	bitrd 526	. . 3 |- (ph -> (ph <-> th))
10	9	ibi 590	. 2 |- (ph -> th)
11		elimhyp2v.5	. . 3 |- ta
12		iffalse 2357	. . . . . 6 |- (-. ph -> if(ph, A, C) = C)
13	12	eqcomd 1472	. . . . 5 |- (-. ph -> C = if(ph, A, C))
14		elimhyp2v.3	. . . . 5 |- (C = if(ph, A, C) -> (ta <-> et))
15	13, 14	syl 10	. . . 4 |- (-. ph -> (ta <-> et))
16		iffalse 2357	. . . . . 6 |- (-. ph -> if(ph, B, D) = D)
17	16	eqcomd 1472	. . . . 5 |- (-. ph -> D = if(ph, B, D))
18		elimhyp2v.4	. . . . 5 |- (D = if(ph, B, D) -> (et <-> th))
19	17, 18	syl 10	. . . 4 |- (-. ph -> (et <-> th))
20	15, 19	bitrd 526	. . 3 |- (-. ph -> (ta <-> th))
21	11, 20	mpbii 193	. 2 |- (-. ph -> th)
22	10, 21	pm2.61i 126	1 |- th

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 953  ifcif 2351

This theorem is referenced by:  bcpasc2t 6906  cvgcmp3cetlem1 7124  cvgcmp3cetlem2 7125  hlimcau 9028  omls 9161  osumlem8 9502

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

Copyright terms: Public domain