Theorem ssrexf 3231

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 3162	. . 3 |- F/ x A C_ B
4		ssel 3163	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
5	4	anim1d 336	. . 3 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	3, 5	eximd 1622	. 2 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ph ) ) )
7		df-rex 2473	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2473	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3imtr4g 205	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1502   e. wcel 2159   F/_wnfc 2318   E.wrex 2468   C_ wss 3143

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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-in 3149  df-ss 3156

This theorem is referenced by: (None)