Theorem eqer 4329

Description: Equivalence relation involving equality of dependent classes A(x) and B(y).

Hypotheses

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

Assertion

Ref	Expression
eqer	|- Er R

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

Proof of Theorem eqer

Step	Hyp	Ref	Expression
1		id 59	. . . 4 |- ([_z / x]_A = [_w / x]_A -> [_z / x]_A = [_w / x]_A)
2	1	eqcomd 1527	. . 3 |- ([_z / x]_A = [_w / x]_A -> [_w / x]_A = [_z / x]_A)
3		eqer.1	. . . 4 |- (x = y -> A = B)
4		eqer.2	. . . 4 |- R = {<.x, y>. | A = B}
5	3, 4	eqerlem 4328	. . 3 |- (zRw <-> [_z / x]_A = [_w / x]_A)
6	3, 4	eqerlem 4328	. . 3 |- (wRz <-> [_w / x]_A = [_z / x]_A)
7	2, 5, 6	3imtr4i 226	. 2 |- (zRw -> wRz)
8		eqtr 1539	. . 3 |- (([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A) -> [_z / x]_A = [_v / x]_A)
9	3, 4	eqerlem 4328	. . . 4 |- (wRv <-> [_w / x]_A = [_v / x]_A)
10	5, 9	anbi12i 493	. . 3 |- ((zRw /\ wRv) <-> ([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A))
11	3, 4	eqerlem 4328	. . 3 |- (zRv <-> [_z / x]_A = [_v / x]_A)
12	8, 10, 11	3imtr4i 226	. 2 |- ((zRw /\ wRv) -> zRv)
13	7, 12	ster 4326	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 230   = wceq 997  [_csb 2051   class class class wbr 2674  {copab 2721  Er wer 4316

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-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  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  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-br 2675  df-opab 2722  df-cnv 3243  df-co 3244  df-er 4319

Copyright terms: Public domain