Theorem elimhyp3v 2396

Description: Eliminate a hypothesis containing 3 class variables.

Hypotheses

Ref	Expression
elimhyp3v.1	|- (A = if(ph, A, D) -> (ph <-> ch))
elimhyp3v.2	|- (B = if(ph, B, R) -> (ch <-> th))
elimhyp3v.3	|- (C = if(ph, C, S) -> (th <-> ta))
elimhyp3v.4	|- (D = if(ph, A, D) -> (et <-> ze))
elimhyp3v.5	|- (R = if(ph, B, R) -> (ze <-> si))
elimhyp3v.6	|- (S = if(ph, C, S) -> (si <-> ta))
elimhyp3v.7	|- et

Assertion

Ref	Expression
elimhyp3v	|- ta

Proof of Theorem elimhyp3v

Step	Hyp	Ref	Expression
1		iftrue 2370	. . . . . 6 |- (ph -> if(ph, A, D) = A)
2	1	eqcomd 1483	. . . . 5 |- (ph -> A = if(ph, A, D))
3		elimhyp3v.1	. . . . 5 |- (A = if(ph, A, D) -> (ph <-> ch))
4	2, 3	syl 10	. . . 4 |- (ph -> (ph <-> ch))
5		iftrue 2370	. . . . . 6 |- (ph -> if(ph, B, R) = B)
6	5	eqcomd 1483	. . . . 5 |- (ph -> B = if(ph, B, R))
7		elimhyp3v.2	. . . . 5 |- (B = if(ph, B, R) -> (ch <-> th))
8	6, 7	syl 10	. . . 4 |- (ph -> (ch <-> th))
9		iftrue 2370	. . . . . 6 |- (ph -> if(ph, C, S) = C)
10	9	eqcomd 1483	. . . . 5 |- (ph -> C = if(ph, C, S))
11		elimhyp3v.3	. . . . 5 |- (C = if(ph, C, S) -> (th <-> ta))
12	10, 11	syl 10	. . . 4 |- (ph -> (th <-> ta))
13	4, 8, 12	3bitrd 546	. . 3 |- (ph -> (ph <-> ta))
14	13	ibi 594	. 2 |- (ph -> ta)
15		elimhyp3v.7	. . 3 |- et
16		iffalse 2371	. . . . . 6 |- (-. ph -> if(ph, A, D) = D)
17	16	eqcomd 1483	. . . . 5 |- (-. ph -> D = if(ph, A, D))
18		elimhyp3v.4	. . . . 5 |- (D = if(ph, A, D) -> (et <-> ze))
19	17, 18	syl 10	. . . 4 |- (-. ph -> (et <-> ze))
20		iffalse 2371	. . . . . 6 |- (-. ph -> if(ph, B, R) = R)
21	20	eqcomd 1483	. . . . 5 |- (-. ph -> R = if(ph, B, R))
22		elimhyp3v.5	. . . . 5 |- (R = if(ph, B, R) -> (ze <-> si))
23	21, 22	syl 10	. . . 4 |- (-. ph -> (ze <-> si))
24		iffalse 2371	. . . . . 6 |- (-. ph -> if(ph, C, S) = S)
25	24	eqcomd 1483	. . . . 5 |- (-. ph -> S = if(ph, C, S))
26		elimhyp3v.6	. . . . 5 |- (S = if(ph, C, S) -> (si <-> ta))
27	25, 26	syl 10	. . . 4 |- (-. ph -> (si <-> ta))
28	19, 23, 27	3bitrd 546	. . 3 |- (-. ph -> (et <-> ta))
29	15, 28	mpbii 193	. 2 |- (-. ph -> ta)
30	14, 29	pm2.61i 126	1 |- ta

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 958  ifcif 2365

This theorem is referenced by:  climuni 7099  hlimuni 9104  projlem7 9187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-if 2366

Copyright terms: Public domain