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Theorem eqer 7363
 Description: Equivalence relation involving equality of dependent classes A(x) and ( ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eqer.1
eqer.2
Assertion
Ref Expression
eqer
Distinct variable groups:   ,   ,   ,

Proof of Theorem eqer
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . . . 5
21relopabi 5133 . . . 4
32a1i 11 . . 3
4 id 22 . . . . . 6
54eqcomd 2465 . . . . 5
6 eqer.1 . . . . . 6
76, 1eqerlem 7362 . . . . 5
86, 1eqerlem 7362 . . . . 5
95, 7, 83imtr4i 266 . . . 4
11 eqtr 2483 . . . . 5
126, 1eqerlem 7362 . . . . . 6
137, 12anbi12i 697 . . . . 5
146, 1eqerlem 7362 . . . . 5
1511, 13, 143imtr4i 266 . . . 4
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395   wtru 1396  e.wcel 1818   cvv 3109  [_csb 3434   class class class wbr 4452  {copab 4509  Relwrel 5009  Erwer 7327 This theorem is referenced by:  ider  7364  frgpuplem  16790  fneer  30171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-er 7330