Theorem keephyp2v 2368

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 2354).

Hypotheses

Ref	Expression
keephyp2v.1	|- (A = if(ph, A, C) -> (ps <-> ch))
keephyp2v.2	|- (B = if(ph, B, D) -> (ch <-> th))
keephyp2v.3	|- (C = if(ph, A, C) -> (ta <-> et))
keephyp2v.4	|- (D = if(ph, B, D) -> (et <-> th))
keephyp2v.5	|- ps
keephyp2v.6	|- ta

Assertion

Ref	Expression
keephyp2v	|- th

Proof of Theorem keephyp2v

Step	Hyp	Ref	Expression
1		keephyp2v.5	. . 3 |- ps
2		iftrue 2337	. . . . . 6 |- (ph -> if(ph, A, C) = A)
3	2	eqcomd 1456	. . . . 5 |- (ph -> A = if(ph, A, C))
4		keephyp2v.1	. . . . 5 |- (A = if(ph, A, C) -> (ps <-> ch))
5	3, 4	syl 10	. . . 4 |- (ph -> (ps <-> ch))
6		iftrue 2337	. . . . . 6 |- (ph -> if(ph, B, D) = B)
7	6	eqcomd 1456	. . . . 5 |- (ph -> B = if(ph, B, D))
8		keephyp2v.2	. . . . 5 |- (B = if(ph, B, D) -> (ch <-> th))
9	7, 8	syl 10	. . . 4 |- (ph -> (ch <-> th))
10	5, 9	bitrd 526	. . 3 |- (ph -> (ps <-> th))
11	1, 10	mpbii 193	. 2 |- (ph -> th)
12		keephyp2v.6	. . 3 |- ta
13		iffalse 2338	. . . . . 6 |- (-. ph -> if(ph, A, C) = C)
14	13	eqcomd 1456	. . . . 5 |- (-. ph -> C = if(ph, A, C))
15		keephyp2v.3	. . . . 5 |- (C = if(ph, A, C) -> (ta <-> et))
16	14, 15	syl 10	. . . 4 |- (-. ph -> (ta <-> et))
17		iffalse 2338	. . . . . 6 |- (-. ph -> if(ph, B, D) = D)
18	17	eqcomd 1456	. . . . 5 |- (-. ph -> D = if(ph, B, D))
19		keephyp2v.4	. . . . 5 |- (D = if(ph, B, D) -> (et <-> th))
20	18, 19	syl 10	. . . 4 |- (-. ph -> (et <-> th))
21	16, 20	bitrd 526	. . 3 |- (-. ph -> (ta <-> th))
22	12, 21	mpbii 193	. 2 |- (-. ph -> th)
23	11, 22	pm2.61i 126	1 |- th

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 1099  ifcif 2332

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-if 2333

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