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Theorem bj-intabssel 11644
Description: Version of intss1 3703 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
StepHypRef Expression
1 bj-intabssel.nf . . 3  |-  F/_ x A
21nfsbc1 2857 . . 3  |-  F/ x [. A  /  x ]. ph
3 sbceq1a 2849 . . 3  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
41, 2, 3elabgf 2758 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  [. A  /  x ]. ph ) )
5 intss1 3703 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl6bir 162 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  ->  |^| { x  |  ph }  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   {cab 2074   F/_wnfc 2215   [.wsbc 2840    C_ wss 2999   |^|cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-in 3005  df-ss 3012  df-int 3689
This theorem is referenced by: (None)
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