Theorem eqer 4271

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 1480	. . 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 4270	. . 3 |- (zRw <-> [_z / x]_A = [_w / x]_A)
6	3, 4	eqerlem 4270	. . 3 |- (wRz <-> [_w / x]_A = [_z / x]_A)
7	2, 5, 6	3imtr4 219	. 2 |- (zRw -> wRz)
8		eqtrt 1492	. . 3 |- (([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A) -> [_z / x]_A = [_v / x]_A)
9	3, 4	eqerlem 4270	. . . 4 |- (wRv <-> [_w / x]_A = [_v / x]_A)
10	5, 9	anbi12i 482	. . 3 |- ((zRw /\ wRv) <-> ([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A))
11	3, 4	eqerlem 4270	. . 3 |- (zRv <-> [_z / x]_A = [_v / x]_A)
12	8, 10, 11	3imtr4 219	. 2 |- ((zRw /\ wRv) -> zRv)
13	7, 12	ster 4268	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956  [_csb 2001   class class class wbr 2619  {copab 2666  Er wer 4258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-co 3187  df-er 4261

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