Theorem bj-intabssel 14894

Description: Version of intss1 3871 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 2992	. . 3 |- F/ x [. A / x ]. ph
3		sbceq1a 2984	. . 3 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
4	1, 2, 3	elabgf 2891	. 2 |- ( A e. V -> ( A e. { x | ph } <-> [. A / x ]. ph ) )
5		intss1 3871	. 2 |- ( A e. { x | ph } -> |^| { x | ph } C_ A )
6	4, 5	syl6bir 164	1 |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Syntax hints:   -> wi 4   e. wcel 2158   {cab 2173   F/_wnfc 2316   [.wsbc 2974   C_ wss 3141   |^|cint 3856

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975  df-in 3147  df-ss 3154  df-int 3857

This theorem is referenced by: (None)