Theorem rexex 2596

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 2534	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1653	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ph )
4	1, 3	sylbi 121	1 |- ( E. x e. A ph -> E. x ph )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534

This theorem is used by:  reu3  3016  rmo2i  3143  dffo5  5857  halfnq  7778  nsmallnq  7780  0npr  7850  genpml  7884  genpmu  7885  ltexprlemm  7967  ltexprlemloc  7974  dedekindeulemlub  15721  dedekindicclemlub  15730