Theorem rexss 3007

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 2939	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 374	. . . 4 |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	anbi1d 438	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. B /\ x e. A ) /\ ph ) ) )
4		anass 381	. . 3 |- ( ( ( x e. B /\ x e. A ) /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) )
5	3, 4	syl6bb 185	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) ) )
6	5	rexbidv2 2329	1 |- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   E.wrex 2307   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-in 2924  df-ss 2931

This theorem is referenced by:  1idprl  6688  1idpru  6689  ltexprlemm  6698