Theorem eqer 7363

Description: Equivalence relation involving equality of dependent classes A(x) and (). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
eqer.1
eqer.2

Assertion

Ref	Expression
eqer

Distinct variable groups:   ,   ,   ,

Proof of Theorem eqer

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . . . 5
2	1	relopabi 5133	. . . 4
3	2	a1i 11	. . 3
4		id 22	. . . . . 6
5	4	eqcomd 2465	. . . . 5
6		eqer.1	. . . . . 6
7	6, 1	eqerlem 7362	. . . . 5
8	6, 1	eqerlem 7362	. . . . 5
9	5, 7, 8	3imtr4i 266	. . . 4
10	9	adantl 466	. . 3
11		eqtr 2483	. . . . 5
12	6, 1	eqerlem 7362	. . . . . 6
13	7, 12	anbi12i 697	. . . . 5
14	6, 1	eqerlem 7362	. . . . 5
15	11, 13, 14	3imtr4i 266	. . . 4
16	15	adantl 466	. . 3
17		vex 3112	. . . . 5
18		eqid 2457	. . . . . 6
19	6, 1	eqerlem 7362	. . . . . 6
20	18, 19	mpbir 209	. . . . 5
21	17, 20	2th 239	. . . 4
22	21	a1i 11	. . 3
23	3, 10, 16, 22	iserd 7356	. 2
24	23	trud 1404	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  wtru 1396  e.wcel 1818  cvv 3109  [_csb 3434   class class class wbr 4452  {copab 4509  Relwrel 5009  Erwer 7327

This theorem is referenced by:  ider  7364  frgpuplem  16790  fneer  30171

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-er 7330