Theorem reximdva0 3562

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 3560	. . 3 |- (A =/= (/) <-> E.x x e. A)
2		reximdva0.1	. . . . . . 7 |- ((ph /\ x e. A) -> ps)
3	2	ex 423	. . . . . 6 |- (ph -> (x e. A -> ps))
4	3	ancld 536	. . . . 5 |- (ph -> (x e. A -> (x e. A /\ ps)))
5	4	eximdv 1622	. . . 4 |- (ph -> (E.x x e. A -> E.x(x e. A /\ ps)))
6	5	imp 418	. . 3 |- ((ph /\ E.x x e. A) -> E.x(x e. A /\ ps))
7	1, 6	sylan2b 461	. 2 |- ((ph /\ A =/= (/)) -> E.x(x e. A /\ ps))
8		df-rex 2621	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
9	7, 8	sylibr 203	1 |- ((ph /\ A =/= (/)) -> E.x e. A ps)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   e. wcel 1710   =/= wne 2517  E.wrex 2616  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)