Theorem rabssf 39302

Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabssf.1	|- F/_ x B

Assertion

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

Proof of Theorem rabssf

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq1i 3629	. 2 |- ( { x e. A | ph } C_ B <-> { x | ( x e. A /\ ph ) } C_ B )
3		rabssf.1	. . 3 |- F/_ x B
4	3	abssf 39295	. 2 |- ( { x | ( x e. A /\ ph ) } C_ B <-> A. x ( ( x e. A /\ ph ) -> x e. B ) )
5		impexp 462	. . . 4 |- ( ( ( x e. A /\ ph ) -> x e. B ) <-> ( x e. A -> ( ph -> x e. B ) ) )
6	5	albii 1747	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
7		df-ral 2917	. . 3 |- ( A. x e. A ( ph -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
8	6, 7	bitr4i 267	. 2 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x e. A ( ph -> x e. B ) )
9	2, 4, 8	3bitri 286	1 |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) )

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:  rabssd  39332  supminfxr2  39699  preimageiingt  40930  preimaleiinlt  40931  smfmullem4  41001