Theorem bj-intabssel 16153

Description: Version of intss1 3938 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

Hypothesis

Ref	Expression
bj-intabssel.nf	|- F/_ x A

Assertion

Ref	Expression
bj-intabssel	|- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Proof of Theorem bj-intabssel

Step	Hyp	Ref	Expression
1		bj-intabssel.nf	. . 3 |- F/_ x A
2	1	nfsbc1 3046	. . 3 |- F/ x [. A / x ]. ph
3		sbceq1a 3038	. . 3 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
4	1, 2, 3	elabgf 2945	. 2 |- ( A e. V -> ( A e. { x | ph } <-> [. A / x ]. ph ) )
5		intss1 3938	. 2 |- ( A e. { x | ph } -> |^| { x | ph } C_ A )
6	4, 5	biimtrrdi 164	1 |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Syntax hints:   -> wi 4   e. wcel 2200   {cab 2215   F/_wnfc 2359   [.wsbc 3028   C_ wss 3197   |^|cint 3923

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-in 3203  df-ss 3210  df-int 3924

This theorem is referenced by: (None)