Theorem rexex 2503

Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
rexex	|- ( E. x e. A ph -> E. x ph )

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2441	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 109	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1580	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ph )
4	1, 3	sylbi 120	1 |- ( E. x e. A ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1472   e. wcel 2128   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-rex 2441

This theorem is referenced by:  reu3  2902  rmo2i  3027  dffo5  5616  halfnq  7331  nsmallnq  7333  0npr  7403  genpml  7437  genpmu  7438  ltexprlemm  7520  ltexprlemloc  7527  dedekindeulemlub  12998  dedekindicclemlub  13007