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Theorem r19.41vv 23923
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. Version 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 2821 . . 3  |-  ( E. y  e.  B  (
ph  /\  ps )  <->  ( E. y  e.  B  ph 
/\  ps ) )
21rexbii 2691 . 2  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  E. x  e.  A  ( E. y  e.  B  ph  /\  ps ) )
3 r19.41v 2821 . 2  |-  ( E. x  e.  A  ( E. y  e.  B  ph 
/\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  ps ) )
42, 3bitri 241 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 set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wrex 2667
This theorem is referenced by:  dya2iocnrect  24584  usg2spot2nb  28168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-rex 2672
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