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Theorem rexcom 2621
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 2299 . 2  |-  F/_ y A
2 nfcv 2299 . 2  |-  F/_ x B
31, 2rexcomf 2619 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 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  rexcom13  2622  rexcom4  2735  iuncom  3857  xpiundi  4646  addcomprg  7500  mulcomprg  7502  ltexprlemm  7522  caucvgprprlemexbt  7628  suplocexprlemml  7638  suplocexprlemmu  7640  qmulz  9538  elpq  9563  caubnd2  11028  sqrt2irr  12052  pythagtriplem19  12172
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