Theorem reximdva0 3933

Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)

Hypothesis

Ref	Expression
reximdva0.1	|- ( ( ph /\ x e. A ) -> ps )

Assertion

Ref	Expression
reximdva0	|- ( ( ph /\ A =/= (/) ) -> E. x e. A ps )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem reximdva0

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( A =/= (/) <-> E. x x e. A )
2		reximdva0.1	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ps )
3	2	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> ps ) )
4	3	ancld 576	. . . . 5 |- ( ph -> ( x e. A -> ( x e. A /\ ps ) ) )
5	4	eximdv 1846	. . . 4 |- ( ph -> ( E. x x e. A -> E. x ( x e. A /\ ps ) ) )
6	5	imp 445	. . 3 |- ( ( ph /\ E. x x e. A ) -> E. x ( x e. A /\ ps ) )
7	1, 6	sylan2b 492	. 2 |- ( ( ph /\ A =/= (/) ) -> E. x ( x e. A /\ ps ) )
8		df-rex 2918	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
9	7, 8	sylibr 224	1 |- ( ( ph /\ A =/= (/) ) -> E. x e. A ps )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  n0snor2el  4364  hashgt12el  13210  refun0  21318  cstucnd  22088  supxrnemnf  29534  kerunit  29823  elpaddn0  35086