Theorem rexex 2674

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

Assertion

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

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2621	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		simpr 447	. . 3 |- ((x e. A /\ ph) -> ph)
3	2	eximi 1576	. 2 |- (E.x(x e. A /\ ph) -> E.xph)
4	1, 3	sylbi 187	1 |- (E.x e. A ph -> E.xph)

This proof depends on syntax axioms:   -> wi 4   /\ wa 358  E.wex 1541   e. wcel 1710  E.wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-rex 2621

This theorem is used by:  reu3  3027  rmo2i  3133  dffo5  5425