Theorem keephyp3v 3029

Description: Keep a hypothesis containing 3 class variables.

Hypotheses

Ref	Expression
keephyp3v.1	|- (A = if(ph, A, D) -> (rh <-> ch))
keephyp3v.2	|- (B = if(ph, B, R) -> (ch <-> th))
keephyp3v.3	|- (C = if(ph, C, S) -> (th <-> ta))
keephyp3v.4	|- (D = if(ph, A, D) -> (et <-> ze))
keephyp3v.5	|- (R = if(ph, B, R) -> (ze <-> si))
keephyp3v.6	|- (S = if(ph, C, S) -> (si <-> ta))
keephyp3v.7	|- rh
keephyp3v.8	|- et

Assertion

Ref	Expression
keephyp3v	|- ta

Proof of Theorem keephyp3v

Step	Hyp	Ref	Expression
1		keephyp3v.7	. . 3 |- rh
2		iftrue 2989	. . . . . 6 |- (ph -> if(ph, A, D) = A)
3	2	eqcomd 1889	. . . . 5 |- (ph -> A = if(ph, A, D))
4		keephyp3v.1	. . . . 5 |- (A = if(ph, A, D) -> (rh <-> ch))
5	3, 4	syl 12	. . . 4 |- (ph -> (rh <-> ch))
6		iftrue 2989	. . . . . 6 |- (ph -> if(ph, B, R) = B)
7	6	eqcomd 1889	. . . . 5 |- (ph -> B = if(ph, B, R))
8		keephyp3v.2	. . . . 5 |- (B = if(ph, B, R) -> (ch <-> th))
9	7, 8	syl 12	. . . 4 |- (ph -> (ch <-> th))
10		iftrue 2989	. . . . . 6 |- (ph -> if(ph, C, S) = C)
11	10	eqcomd 1889	. . . . 5 |- (ph -> C = if(ph, C, S))
12		keephyp3v.3	. . . . 5 |- (C = if(ph, C, S) -> (th <-> ta))
13	11, 12	syl 12	. . . 4 |- (ph -> (th <-> ta))
14	5, 9, 13	3bitrd 603	. . 3 |- (ph -> (rh <-> ta))
15	1, 14	mpbii 210	. 2 |- (ph -> ta)
16		keephyp3v.8	. . 3 |- et
17		iffalse 2991	. . . . . 6 |- (-. ph -> if(ph, A, D) = D)
18	17	eqcomd 1889	. . . . 5 |- (-. ph -> D = if(ph, A, D))
19		keephyp3v.4	. . . . 5 |- (D = if(ph, A, D) -> (et <-> ze))
20	18, 19	syl 12	. . . 4 |- (-. ph -> (et <-> ze))
21		iffalse 2991	. . . . . 6 |- (-. ph -> if(ph, B, R) = R)
22	21	eqcomd 1889	. . . . 5 |- (-. ph -> R = if(ph, B, R))
23		keephyp3v.5	. . . . 5 |- (R = if(ph, B, R) -> (ze <-> si))
24	22, 23	syl 12	. . . 4 |- (-. ph -> (ze <-> si))
25		iffalse 2991	. . . . . 6 |- (-. ph -> if(ph, C, S) = S)
26	25	eqcomd 1889	. . . . 5 |- (-. ph -> S = if(ph, C, S))
27		keephyp3v.6	. . . . 5 |- (S = if(ph, C, S) -> (si <-> ta))
28	26, 27	syl 12	. . . 4 |- (-. ph -> (si <-> ta))
29	20, 24, 28	3bitrd 603	. . 3 |- (-. ph -> (et <-> ta))
30	16, 29	mpbii 210	. 2 |- (-. ph -> ta)
31	15, 30	pm2.61i 140	1 |- ta

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298  ifcif 2982

This theorem is referenced by:  projlem7 10825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

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