Theorem ssrexf 3209

Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
ssrexf.1	|- F/_ x A
ssrexf.2	|- F/_ x B

Assertion

Ref	Expression
ssrexf	|- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Proof of Theorem ssrexf

Step	Hyp	Ref	Expression
1		ssrexf.1	. . . 4 |- F/_ x A
2		ssrexf.2	. . . 4 |- F/_ x B
3	1, 2	nfss 3140	. . 3 |- F/ x A C_ B
4		ssel 3141	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
5	4	anim1d 334	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	3, 5	eximd 1605	. 2 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ph ) ) )
7		df-rex 2454	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2454	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3imtr4g 204	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   e. wcel 2141   F/_wnfc 2299   E.wrex 2449   C_ wss 3121

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-in 3127  df-ss 3134

This theorem is referenced by: (None)