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Theorem rexcom4a 2632
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Assertion
Ref Expression
rexcom4a  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Distinct variable groups:    x, A    x, y    ph, x
Allowed substitution hints:    ph( y)    ps( x, y)    A( y)

Proof of Theorem rexcom4a
StepHypRef Expression
1 rexcom4 2631 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. x E. y  e.  A  ( ph  /\ 
ps ) )
2 19.42v 1829 . . 3  |-  ( E. x ( ph  /\  ps )  <->  ( ph  /\  E. x ps ) )
32rexbii 2378 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
41, 3bitr3i 184 1  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422   E.wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612
This theorem is referenced by:  rexcom4b  2633
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