Theorem rabssrabd 3315

Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.)

Hypotheses

Ref	Expression
rabssrabd.1	|- ( ph -> A C_ B )
rabssrabd.2	|- ( ( ph /\ ps /\ x e. A ) -> ch )

Assertion

Ref	Expression
rabssrabd	|- ( ph -> { x e. A | ps } C_ { x e. B | ch } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem rabssrabd

Step	Hyp	Ref	Expression
1		3anan32 1016	. . . . 5 |- ( ( ph /\ ps /\ x e. A ) <-> ( ( ph /\ x e. A ) /\ ps ) )
2		rabssrabd.2	. . . . 5 |- ( ( ph /\ ps /\ x e. A ) -> ch )
3	1, 2	sylbir 135	. . . 4 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
4	3	ex 115	. . 3 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
5	4	ss2rabdv 3309	. 2 |- ( ph -> { x e. A | ps } C_ { x e. A | ch } )
6		rabssrabd.1	. . 3 |- ( ph -> A C_ B )
7		rabss2 3311	. . 3 |- ( A C_ B -> { x e. A | ch } C_ { x e. B | ch } )
8	6, 7	syl 14	. 2 |- ( ph -> { x e. A | ch } C_ { x e. B | ch } )
9	5, 8	sstrd 3238	1 |- ( ph -> { x e. A | ps } C_ { x e. B | ch } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   {crab 2515   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-3an 1007  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-in 3207  df-ss 3214

This theorem is referenced by:  suppfnss  6435