Theorem rexex 1685

Description: Restricted existence implies existence.

Assertion

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

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 1642	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		pm3.27 323	. . 3 |- ((x e. A /\ ph) -> ph)
3	2	19.22i 1036	. 2 |- (E.x(x e. A /\ ph) -> E.xph)
4	1, 3	sylbi 199	1 |- (E.x e. A ph -> E.xph)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  E.wex 977  E.wrex 1638

This theorem is referenced by:  reu6 1922  dffo5 3806  ivthlem6 7221  ivthlem7 7222  ivthlem6OLD 7230  ivthlem7OLD 7231

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-rex 1642

Copyright terms: Public domain