Theorem rexex 2477

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 2420	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 109	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1579	. 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 1468   e. wcel 1480   E.wrex 2415

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-rex 2420

This theorem is referenced by:  reu3  2869  rmo2i  2994  dffo5  5562  halfnq  7212  nsmallnq  7214  0npr  7284  genpml  7318  genpmu  7319  ltexprlemm  7401  ltexprlemloc  7408  dedekindeulemlub  12756  dedekindicclemlub  12765