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Theorem ralcom 2652
Description: Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
ralcom  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
Distinct variable groups:    x, y    x, B    y, A
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem ralcom
StepHypRef Expression
1 nfcv 2331 . 2  |-  F/_ y A
2 nfcv 2331 . 2  |-  F/_ x B
31, 2ralcomf 2650 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wral 2467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472
This theorem is referenced by:  ralrot3  2654  ralcom4  2773  ssint  3874  issod  4333  reusv3  4474  cnvpom  5185  cnvsom  5186  fununi  5298  isocnv2  5828  dfsmo2  6305  ixpiinm  6741  rexfiuz  11015  isnsg2  13107  opprsubrngg  13518  rmodislmodlem  13626  rmodislmod  13627  tgss2  13962  cnmptcom  14181
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