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Theorem rexcom 2532
Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
rexcom  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. 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 rexcom
StepHypRef Expression
1 nfcv 2229 . 2  |-  F/_ y A
2 nfcv 2229 . 2  |-  F/_ x B
31, 2rexcomf 2530 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wrex 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366
This theorem is referenced by:  rexcom13  2533  rexcom4  2643  iuncom  3742  xpiundi  4509  addcomprg  7198  mulcomprg  7200  ltexprlemm  7220  caucvgprprlemexbt  7326  qmulz  9169  caubnd2  10611  sqrt2irr  11480
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