Theorem ssrexf 3665

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 3596	. . 3 |- F/ x A C_ B
4		ssel 3597	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
5	4	anim1d 588	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	3, 5	eximd 2085	. 2 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ph ) ) )
7		df-rex 2918	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2918	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3imtr4g 285	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-in 3581  df-ss 3588

This theorem is referenced by:  iunxdif3  4606  stoweidlem34  40251