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Theorem 2ralbii 2569
Description: Inference adding 2 restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
ralbii.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
2ralbii  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x  e.  A  A. y  e.  B  ps )

Proof of Theorem 2ralbii
StepHypRef Expression
1 ralbii.1 . . 3  |-  ( ph  <->  ps )
21ralbii 2567 . 2  |-  ( A. y  e.  B  ph  <->  A. y  e.  B  ps )
32ralbii 2567 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x  e.  A  A. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wral 2543
This theorem is referenced by:  cnvso  5214  fununi  5316  isocnv2  5828  tpossym  6266  sorpss  6282  dford2  7321  ispos2  14082  odulatb  14247  issubm  14425  cntzrec  14809  oppgsubm  14835  opprirred  15484  opprsubrg  15566  isbasis2g  16686  ist0-3  17073  isfbas2  17530  dfadj2  22465  adjval2  22471  cnlnadjeui  22657  adjbdln  22663  rmo4f  23180  iccllyscon  23781  dfso3  24074  elpotr  24137  dfon2  24148  r19.26-2a  24934  dfdir2  25291  trooo  25394  issrc  25867  propsrc  25868  dfcon2OLD  26253  f1opr  26391  fphpd  26899  isdomn3  27523  2reu4a  27967  ordelordALT  28301  isltrn2N  30309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-nf 1532  df-ral 2548
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