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Theorem raleqi 2740
Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
raleq1i.1  |-  A  =  B
Assertion
Ref Expression
raleqi  |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem raleqi
StepHypRef Expression
1 raleq1i.1 . 2  |-  A  =  B
2 raleq 2736 . 2  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
31, 2ax-mp 8 1  |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   A.wral 2543
This theorem is referenced by:  ralrab2  2931  ralprg  3682  raltpg  3684  ralxp  4827  ralrnmpt2  5958  ovmptss  6200  ixpfi2  7154  dffi3  7184  dfoi  7226  fseqenlem1  7651  kmlem12  7787  fzprval  10844  fztpval  10845  hashbc  11391  2prm  12774  prmreclem2  12964  xpsfrnel  13465  xpsle  13483  gsumwspan  14468  basdif0  16691  ordtbaslem  16918  ptbasfi  17276  ptcnplem  17315  ptrescn  17333  flftg  17691  minveclem1  18788  minveclem3b  18792  minveclem6  18798  iblcnlem1  19142  ellimc2  19227  ftalem3  20312  dchreq  20497  pntlem3  20758  elghom  21030  minvecolem1  21453  minvecolem5  21460  minvecolem6  21461  cdj3lem3b  23020  cnvref  25065  repcpwti  25161  filnetlem4  26330  iscrngo2  26623  fnwe2lem2  27148  islinds2  27283  psgnunilem3  27419  tendoset  30948
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548
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