Theorem eqerOLD 7778

Description: Obsolete proof of eqer 7777 as of 1-May-2021. Equivalence relation involving equality of dependent classes A ( x ) and B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eqer.1	|- ( x = y -> A = B )
eqer.2	|- R = { <. x , y >. | A = B }

Assertion

Ref	Expression
eqerOLD	|- R Er _V

Distinct variable groups:   x, y   y, A   x, B

Allowed substitution hints:   A( x)   B( y)   R( x, y)

Proof of Theorem eqerOLD

Dummy variables w z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . . . 5 |- R = { <. x , y >. | A = B }
2	1	relopabi 5245	. . . 4 |- Rel R
3	2	a1i 11	. . 3 |- ( T. -> Rel R )
4		id 22	. . . . . 6 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ z / x ]_ A = [_ w / x ]_ A )
5	4	eqcomd 2628	. . . . 5 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ w / x ]_ A = [_ z / x ]_ A )
6		eqer.1	. . . . . 6 |- ( x = y -> A = B )
7	6, 1	eqerlem 7776	. . . . 5 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
8	6, 1	eqerlem 7776	. . . . 5 |- ( w R z <-> [_ w / x ]_ A = [_ z / x ]_ A )
9	5, 7, 8	3imtr4i 281	. . . 4 |- ( z R w -> w R z )
10	9	adantl 482	. . 3 |- ( ( T. /\ z R w ) -> w R z )
11		eqtr 2641	. . . . 5 |- ( ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) -> [_ z / x ]_ A = [_ v / x ]_ A )
12	6, 1	eqerlem 7776	. . . . . 6 |- ( w R v <-> [_ w / x ]_ A = [_ v / x ]_ A )
13	7, 12	anbi12i 733	. . . . 5 |- ( ( z R w /\ w R v ) <-> ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) )
14	6, 1	eqerlem 7776	. . . . 5 |- ( z R v <-> [_ z / x ]_ A = [_ v / x ]_ A )
15	11, 13, 14	3imtr4i 281	. . . 4 |- ( ( z R w /\ w R v ) -> z R v )
16	15	adantl 482	. . 3 |- ( ( T. /\ ( z R w /\ w R v ) ) -> z R v )
17		vex 3203	. . . . 5 |- z e. _V
18		eqid 2622	. . . . . 6 |- [_ z / x ]_ A = [_ z / x ]_ A
19	6, 1	eqerlem 7776	. . . . . 6 |- ( z R z <-> [_ z / x ]_ A = [_ z / x ]_ A )
20	18, 19	mpbir 221	. . . . 5 |- z R z
21	17, 20	2th 254	. . . 4 |- ( z e. _V <-> z R z )
22	21	a1i 11	. . 3 |- ( T. -> ( z e. _V <-> z R z ) )
23	3, 10, 16, 22	iserd 7768	. 2 |- ( T. -> R Er _V )
24	23	trud 1493	1 |- R Er _V

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   [_csb 3533   class class class wbr 4653   {copab 4712   Rel wrel 5119   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by: (None)