Theorem rexss 3195

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 3122	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 392	. . . 4 |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	anbi1d 461	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. B /\ x e. A ) /\ ph ) ) )
4		anass 399	. . 3 |- ( ( ( x e. B /\ x e. A ) /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) )
5	3, 4	bitrdi 195	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) ) )
6	5	rexbidv2 2460	1 |- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   E.wrex 2436   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-rex 2441  df-in 3108  df-ss 3115

This theorem is referenced by:  1idprl  7510  1idpru  7511  ltexprlemm  7520  suplocexprlemmu  7638  oddnn02np1  11770  oddge22np1  11771  evennn02n  11772  evennn2n  11773