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Theorem r19.41vv 3008
Description: Version of r19.41v 3006 with two quantifiers (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
r19.41vv  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  ps ) )
Distinct variable groups:    ps, x    ps, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem r19.41vv
StepHypRef Expression
1 r19.41v 3006 . . 3  |-  ( E. y  e.  B  (
ph  /\  ps )  <->  ( E. y  e.  B  ph 
/\  ps ) )
21rexbii 2956 . 2  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  E. x  e.  A  ( E. y  e.  B  ph  /\  ps ) )
3 r19.41v 3006 . 2  |-  ( E. x  e.  A  ( E. y  e.  B  ph 
/\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  ps ) )
42, 3bitri 249 1  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-rex 2810
This theorem is referenced by:  genpass  9376  axeuclid  24468  usg2spot2nb  25267  dya2iocnrect  28489  nofulllem5  29706  itg2addnclem3  30308
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