Theorem eqer 4277

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 1483	. . 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 4276	. . 3 |- (zRw <-> [_z / x]_A = [_w / x]_A)
6	3, 4	eqerlem 4276	. . 3 |- (wRz <-> [_w / x]_A = [_z / x]_A)
7	2, 5, 6	3imtr4 219	. 2 |- (zRw -> wRz)
8		eqtrt 1495	. . 3 |- (([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A) -> [_z / x]_A = [_v / x]_A)
9	3, 4	eqerlem 4276	. . . 4 |- (wRv <-> [_w / x]_A = [_v / x]_A)
10	5, 9	anbi12i 484	. . 3 |- ((zRw /\ wRv) <-> ([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A))
11	3, 4	eqerlem 4276	. . 3 |- (zRv <-> [_z / x]_A = [_v / x]_A)
12	8, 10, 11	3imtr4 219	. 2 |- ((zRw /\ wRv) -> zRv)
13	7, 12	ster 4274	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958  [_csb 2004   class class class wbr 2624  {copab 2671  Er wer 4264

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192  df-co 3193  df-er 4267

Copyright terms: Public domain