Theorem elimhyp4v 2445

Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 2435).

Hypotheses

Ref	Expression
elimhyp4v.1	|- (A = if(ph, A, D) -> (ph <-> ch))
elimhyp4v.2	|- (B = if(ph, B, R) -> (ch <-> th))
elimhyp4v.3	|- (C = if(ph, C, S) -> (th <-> ta))
elimhyp4v.4	|- (F = if(ph, F, G) -> (ta <-> ps))
elimhyp4v.5	|- (D = if(ph, A, D) -> (et <-> ze))
elimhyp4v.6	|- (R = if(ph, B, R) -> (ze <-> si))
elimhyp4v.7	|- (S = if(ph, C, S) -> (si <-> rh))
elimhyp4v.8	|- (G = if(ph, F, G) -> (rh <-> ps))
elimhyp4v.9	|- et

Assertion

Ref	Expression
elimhyp4v	|- ps

Proof of Theorem elimhyp4v

Step	Hyp	Ref	Expression
1		iftrue 2418	. . . . . . 7 |- (ph -> if(ph, A, D) = A)
2	1	eqcomd 1527	. . . . . 6 |- (ph -> A = if(ph, A, D))
3		elimhyp4v.1	. . . . . 6 |- (A = if(ph, A, D) -> (ph <-> ch))
4	2, 3	syl 10	. . . . 5 |- (ph -> (ph <-> ch))
5		iftrue 2418	. . . . . . 7 |- (ph -> if(ph, B, R) = B)
6	5	eqcomd 1527	. . . . . 6 |- (ph -> B = if(ph, B, R))
7		elimhyp4v.2	. . . . . 6 |- (B = if(ph, B, R) -> (ch <-> th))
8	6, 7	syl 10	. . . . 5 |- (ph -> (ch <-> th))
9	4, 8	bitrd 539	. . . 4 |- (ph -> (ph <-> th))
10		iftrue 2418	. . . . . 6 |- (ph -> if(ph, C, S) = C)
11	10	eqcomd 1527	. . . . 5 |- (ph -> C = if(ph, C, S))
12		elimhyp4v.3	. . . . 5 |- (C = if(ph, C, S) -> (th <-> ta))
13	11, 12	syl 10	. . . 4 |- (ph -> (th <-> ta))
14		iftrue 2418	. . . . . 6 |- (ph -> if(ph, F, G) = F)
15	14	eqcomd 1527	. . . . 5 |- (ph -> F = if(ph, F, G))
16		elimhyp4v.4	. . . . 5 |- (F = if(ph, F, G) -> (ta <-> ps))
17	15, 16	syl 10	. . . 4 |- (ph -> (ta <-> ps))
18	9, 13, 17	3bitrd 555	. . 3 |- (ph -> (ph <-> ps))
19	18	ibi 603	. 2 |- (ph -> ps)
20		elimhyp4v.9	. . 3 |- et
21		iffalse 2419	. . . . . . 7 |- (-. ph -> if(ph, A, D) = D)
22	21	eqcomd 1527	. . . . . 6 |- (-. ph -> D = if(ph, A, D))
23		elimhyp4v.5	. . . . . 6 |- (D = if(ph, A, D) -> (et <-> ze))
24	22, 23	syl 10	. . . . 5 |- (-. ph -> (et <-> ze))
25		iffalse 2419	. . . . . . 7 |- (-. ph -> if(ph, B, R) = R)
26	25	eqcomd 1527	. . . . . 6 |- (-. ph -> R = if(ph, B, R))
27		elimhyp4v.6	. . . . . 6 |- (R = if(ph, B, R) -> (ze <-> si))
28	26, 27	syl 10	. . . . 5 |- (-. ph -> (ze <-> si))
29	24, 28	bitrd 539	. . . 4 |- (-. ph -> (et <-> si))
30		iffalse 2419	. . . . . 6 |- (-. ph -> if(ph, C, S) = S)
31	30	eqcomd 1527	. . . . 5 |- (-. ph -> S = if(ph, C, S))
32		elimhyp4v.7	. . . . 5 |- (S = if(ph, C, S) -> (si <-> rh))
33	31, 32	syl 10	. . . 4 |- (-. ph -> (si <-> rh))
34		iffalse 2419	. . . . . 6 |- (-. ph -> if(ph, F, G) = G)
35	34	eqcomd 1527	. . . . 5 |- (-. ph -> G = if(ph, F, G))
36		elimhyp4v.8	. . . . 5 |- (G = if(ph, F, G) -> (rh <-> ps))
37	35, 36	syl 10	. . . 4 |- (-. ph -> (rh <-> ps))
38	29, 33, 37	3bitrd 555	. . 3 |- (-. ph -> (et <-> ps))
39	20, 38	mpbii 200	. 2 |- (-. ph -> ps)
40	19, 39	pm2.61i 132	1 |- ps

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   = wceq 997  ifcif 2413

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-if 2414

Copyright terms: Public domain