Theorem bj-intabssel 16490

Description: Version of intss1 3948 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 3050	. . 3 |- F/ x [. A / x ]. ph
3		sbceq1a 3042	. . 3 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
4	1, 2, 3	elabgf 2949	. 2 |- ( A e. V -> ( A e. { x | ph } <-> [. A / x ]. ph ) )
5		intss1 3948	. 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 2202   {cab 2217   F/_wnfc 2362   [.wsbc 3032   C_ wss 3201   |^|cint 3933

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-in 3207  df-ss 3214  df-int 3934

This theorem is referenced by: (None)