MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rsp2e Unicode version

Theorem rsp2e 2606
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 955 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  x  e.  A )
2 rspe 2604 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
323adant1 973 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
4 19.8a 1718 . . 3  |-  ( ( x  e.  A  /\  E. y  e.  B  ph )  ->  E. x ( x  e.  A  /\  E. y  e.  B  ph )
)
51, 3, 4syl2anc 642 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
6 df-rex 2549 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
75, 6sylibr 203 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 358    /\ w3a 934   E.wex 1528    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  intopcoaconb  25540  pell14qrdich  26954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1529  df-rex 2549
  Copyright terms: Public domain W3C validator