Theorem rabssd 39332

Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
rabssd.1	|- F/ x ph
rabssd.2	|- F/_ x B
rabssd.3	|- ( ( ph /\ x e. A /\ ch ) -> x e. B )

Assertion

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

Proof of Theorem rabssd

Step	Hyp	Ref	Expression
1		rabssd.1	. . 3 |- F/ x ph
2		rabssd.3	. . . 4 |- ( ( ph /\ x e. A /\ ch ) -> x e. B )
3	2	3exp 1264	. . 3 |- ( ph -> ( x e. A -> ( ch -> x e. B ) ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A ( ch -> x e. B ) )
5		rabssd.2	. . 3 |- F/_ x B
6	5	rabssf 39302	. 2 |- ( { x e. A | ch } C_ B <-> A. x e. A ( ch -> x e. B ) )
7	4, 6	sylibr 224	1 |- ( ph -> { x e. A | ch } C_ B )

Syntax hints:   -> wi 4   /\ w3a 1037   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   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-3an 1039  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-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by: (None)