Theorem ssrexf 3290

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 3221	. . 3 |- F/ x A C_ B
4		ssel 3222	. . . 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 1661	. 2 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ph ) ) )
7		df-rex 2517	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2517	. 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 1541   e. wcel 2202   F/_wnfc 2362   E.wrex 2512   C_ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-in 3207  df-ss 3214

This theorem is referenced by: (None)