Theorem ssrabf 39298

Description: Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
ssrabf.1	|- F/_ x B
ssrabf.2	|- F/_ x A

Assertion

Ref	Expression
ssrabf	|- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )

Proof of Theorem ssrabf

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq2i 3630	. 2 |- ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3		ssrabf.1	. . 3 |- F/_ x B
4	3	ssabf 39280	. 2 |- ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
5		ssrabf.2	. . . . 5 |- F/_ x A
6	3, 5	dfss3f 3595	. . . 4 |- ( B C_ A <-> A. x e. B x e. A )
7	6	anbi1i 731	. . 3 |- ( ( B C_ A /\ A. x e. B ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
8		r19.26 3064	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
9		df-ral 2917	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
10	7, 8, 9	3bitr2ri 289	. 2 |- ( A. x ( x e. B -> ( x e. A /\ ph ) ) <-> ( B C_ A /\ A. x e. B ph ) )
11	2, 4, 10	3bitri 286	1 |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   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-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:  supminfxr2  39699  pimgtmnf2  40924  smfmullem4  41001  smflimsuplem7  41032