Theorem rexss 3259

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 3186	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 394	. . . 4 |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	anbi1d 465	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. B /\ x e. A ) /\ ph ) ) )
4		anass 401	. . 3 |- ( ( ( x e. B /\ x e. A ) /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) )
5	3, 4	bitrdi 196	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) ) )
6	5	rexbidv2 2508	1 |- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2175   E.wrex 2484   C_ wss 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-rex 2489  df-in 3171  df-ss 3178

This theorem is referenced by:  1idprl  7702  1idpru  7703  ltexprlemm  7712  suplocexprlemmu  7830  oddnn02np1  12162  oddge22np1  12163  evennn02n  12164  evennn2n  12165  2lgslem1a  15536