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Theorem r19.41vv 2879
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 2878 . . 3  |-  ( E. y  e.  B  (
ph  /\  ps )  <->  ( E. y  e.  B  ph 
/\  ps ) )
21rexbii 2745 . 2  |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  E. x  e.  A  ( E. y  e.  B  ph  /\  ps ) )
3 r19.41v 2878 . 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 369   E.wrex 2721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-rex 2726
This theorem is referenced by:  dya2iocnrect  26701  usg2spot2nb  30663
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