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Theorem rsp2e 2923
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
Assertion
Ref Expression
rsp2e  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x  e.  A  E. y  e.  B  ph )

Proof of Theorem rsp2e
StepHypRef Expression
1 3anass 977 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  <->  ( x  e.  A  /\  ( y  e.  B  /\  ph ) ) )
2 rspe 2922 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
32anim2i 569 . . 3  |-  ( ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  -> 
( x  e.  A  /\  E. y  e.  B  ph ) )
4 rspe 2922 . . 3  |-  ( ( x  e.  A  /\  E. y  e.  B  ph )  ->  E. x  e.  A  E. y  e.  B  ph )
53, 4syl 16 . 2  |-  ( ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  ->  E. x  e.  A  E. y  e.  B  ph )
61, 5sylbi 195 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1597  df-rex 2820
This theorem is referenced by:  pell14qrdich  30437
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