Theorem gencbvex 3153

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
gencbvex.1
gencbvex.2
gencbvex.3
gencbvex.4

Assertion

Ref	Expression
gencbvex

Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem gencbvex

Step	Hyp	Ref	Expression
1		excom 1849	. 2
2		gencbvex.1	. . . 4
3		gencbvex.3	. . . . . . 7
4		gencbvex.2	. . . . . . 7
5	3, 4	anbi12d 710	. . . . . 6
6	5	bicomd 201	. . . . 5
7	6	eqcoms 2469	. . . 4
8	2, 7	ceqsexv 3146	. . 3
9	8	exbii 1667	. 2
10		19.41v 1771	. . . 4
11		simpr 461	. . . . 5
12		gencbvex.4	. . . . . . . 8
13		eqcom 2466	. . . . . . . . . . 11
14	13	biimpi 194	. . . . . . . . . 10
15	14	adantl 466	. . . . . . . . 9
16	15	eximi 1656	. . . . . . . 8
17	12, 16	sylbi 195	. . . . . . 7
18	17	adantr 465	. . . . . 6
19	18	ancri 552	. . . . 5
20	11, 19	impbii 188	. . . 4
21	10, 20	bitri 249	. . 3
22	21	exbii 1667	. 2
23	1, 9, 22	3bitr3i 275	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109

This theorem is referenced by:  gencbvex2  3154  gencbval  3155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111