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Theorem rsp2e 2761
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.)
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 simp1 957 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  x  e.  A )
2 rspe 2759 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
323adant1 975 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
4 19.8a 1762 . . 3  |-  ( ( x  e.  A  /\  E. y  e.  B  ph )  ->  E. x ( x  e.  A  /\  E. y  e.  B  ph )
)
51, 3, 4syl2anc 643 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
6 df-rex 2703 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
75, 6sylibr 204 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    e. wcel 1725   E.wrex 2698
This theorem is referenced by:  pell14qrdich  26923
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-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-rex 2703
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