Theorem rabss3d 29351

Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Hypothesis

Ref	Expression
rabss3d.1	|- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B )

Assertion

Ref	Expression
rabss3d	|- ( ph -> { x e. A | ps } C_ { x e. B | ps } )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rabss3d

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x ph
2		nfrab1 3122	. 2 |- F/_ x { x e. A | ps }
3		nfrab1 3122	. 2 |- F/_ x { x e. B | ps }
4		rabss3d.1	. . . . 5 |- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B )
5		simprr 796	. . . . 5 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ps )
6	4, 5	jca 554	. . . 4 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ( x e. B /\ ps ) )
7	6	ex 450	. . 3 |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ps ) ) )
8		rabid 3116	. . 3 |- ( x e. { x e. A | ps } <-> ( x e. A /\ ps ) )
9		rabid 3116	. . 3 |- ( x e. { x e. B | ps } <-> ( x e. B /\ ps ) )
10	7, 8, 9	3imtr4g 285	. 2 |- ( ph -> ( x e. { x e. A | ps } -> x e. { x e. B | ps } ) )
11	1, 2, 3, 10	ssrd 3608	1 |- ( ph -> { x e. A | ps } C_ { x e. B | ps } )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {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-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  xpinpreima2  29953  reprss  30695  reprinfz1  30700