Theorem rexss 3334

Description: Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
rexss	|- (A C_ B -> (E.x e. A ph <-> E.x e. B (x e. A /\ ph)))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   ph(x)

Proof of Theorem rexss

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 |- (A C_ B -> (x e. A -> x e. B))
2	1	pm4.71rd 616	. . . 4 |- (A C_ B -> (x e. A <-> (x e. B /\ x e. A)))
3	2	anbi1d 685	. . 3 |- (A C_ B -> ((x e. A /\ ph) <-> ((x e. B /\ x e. A) /\ ph)))
4		anass 630	. . 3 |- (((x e. B /\ x e. A) /\ ph) <-> (x e. B /\ (x e. A /\ ph)))
5	3, 4	syl6bb 252	. 2 |- (A C_ B -> ((x e. A /\ ph) <-> (x e. B /\ (x e. A /\ ph))))
6	5	rexbidv2 2638	1 |- (A C_ B -> (E.x e. A ph <-> E.x e. B (x e. A /\ ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  E.wrex 2616   C_ wss 3258

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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)