Theorem bj-intabssel 13630

Description: Version of intss1 3838 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 2967	. . 3 |- F/ x [. A / x ]. ph
3		sbceq1a 2959	. . 3 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
4	1, 2, 3	elabgf 2867	. 2 |- ( A e. V -> ( A e. { x | ph } <-> [. A / x ]. ph ) )
5		intss1 3838	. 2 |- ( A e. { x | ph } -> |^| { x | ph } C_ A )
6	4, 5	syl6bir 163	1 |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Syntax hints:   -> wi 4   e. wcel 2136   {cab 2151   F/_wnfc 2294   [.wsbc 2950   C_ wss 3115   |^|cint 3823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-sbc 2951  df-in 3121  df-ss 3128  df-int 3824

This theorem is referenced by: (None)