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Theorem rexcom4a 2968
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 2967 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. x E. y  e.  A  ( ph  /\ 
ps ) )
2 19.42v 1928 . . 3  |-  ( E. x ( ph  /\  ps )  <->  ( ph  /\  E. x ps ) )
32rexbii 2722 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
41, 3bitr3i 243 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:    <-> wb 177    /\ wa 359   E.wex 1550   E.wrex 2698
This theorem is referenced by:  rexcom4b  2969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950
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