Theorem rexex 2422

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 2365	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 108	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1536	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ph )
4	1, 3	sylbi 119	1 |- ( E. x e. A ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1426   e. wcel 1438   E.wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-rex 2365

This theorem is referenced by:  reu3  2805  rmo2i  2929  dffo5  5442  halfnq  6960  nsmallnq  6962  0npr  7032  genpml  7066  genpmu  7067  ltexprlemm  7149  ltexprlemloc  7156