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Theorem 2ralbii 2582
Description: Inference adding two 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 2580 . 2  |-  ( A. y  e.  B  ph  <->  A. y  e.  B  ps )
32ralbii 2580 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 2556
This theorem is referenced by:  cnvso  5230  fununi  5332  isocnv2  5844  tpossym  6282  sorpss  6298  dford2  7337  ispos2  14098  odulatb  14263  issubm  14441  cntzrec  14825  oppgsubm  14851  opprirred  15500  opprsubrg  15582  isbasis2g  16702  ist0-3  17089  isfbas2  17546  dfadj2  22481  adjval2  22487  cnlnadjeui  22673  adjbdln  22679  rmo4f  23196  iccllyscon  23796  dfso3  24089  elpotr  24208  dfon2  24219  r19.26-2a  25037  dfdir2  25394  trooo  25497  issrc  25970  propsrc  25971  dfcon2OLD  26356  f1opr  26494  fphpd  27002  isdomn3  27626  2reu4a  28070  ordelordALT  28600  isltrn2N  30931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-nf 1535  df-ral 2561
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