Theorem bj-intabssel 12996

Description: Version of intss1 3786 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 2926	. . 3 |- F/ x [. A / x ]. ph
3		sbceq1a 2918	. . 3 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
4	1, 2, 3	elabgf 2826	. 2 |- ( A e. V -> ( A e. { x | ph } <-> [. A / x ]. ph ) )
5		intss1 3786	. 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 1480   {cab 2125   F/_wnfc 2268   [.wsbc 2909   C_ wss 3071   |^|cint 3771

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-in 3077  df-ss 3084  df-int 3772

This theorem is referenced by: (None)