Theorem eqer 5329

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 73	. . . 4 |- ([_z / x]_A = [_w / x]_A -> [_z / x]_A = [_w / x]_A)
2	1	eqcomd 1889	. . 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 5328	. . 3 |- (zRw <-> [_z / x]_A = [_w / x]_A)
6	3, 4	eqerlem 5328	. . 3 |- (wRz <-> [_w / x]_A = [_z / x]_A)
7	2, 5, 6	3imtr4i 236	. 2 |- (zRw -> wRz)
8		eqtr 1904	. . 3 |- (([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A) -> [_z / x]_A = [_v / x]_A)
9	3, 4	eqerlem 5328	. . . 4 |- (wRv <-> [_w / x]_A = [_v / x]_A)
10	5, 9	anbi12i 540	. . 3 |- ((zRw /\ wRv) <-> ([_z / x]_A = [_w / x]_A /\ [_w / x]_A = [_v / x]_A))
11	3, 4	eqerlem 5328	. . 3 |- (zRv <-> [_z / x]_A = [_v / x]_A)
12	8, 10, 11	3imtr4i 236	. 2 |- ((zRw /\ wRv) -> zRv)
13	7, 12	ster 5325	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298  [_csb 2540   class class class wbr 3338  {copab 3395  Er wer 5315

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-er 5318

Copyright terms: Public domain