ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rsp2e Unicode version

Theorem rsp2e 2389
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 915 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  x  e.  A )
2 rspe 2387 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
323adant1 933 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
4 19.8a 1498 . . 3  |-  ( ( x  e.  A  /\  E. y  e.  B  ph )  ->  E. x ( x  e.  A  /\  E. y  e.  B  ph )
)
51, 3, 4syl2anc 397 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
6 df-rex 2329 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x
( x  e.  A  /\  E. y  e.  B  ph ) )
75, 6sylibr 141 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 101    /\ w3a 896   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-3an 898  df-rex 2329
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator