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Theorem bj-intabssel 12923
Description: Version of intss1 3756 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 2899 . . 3  |-  F/ x [. A  /  x ]. ph
3 sbceq1a 2891 . . 3  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
41, 2, 3elabgf 2800 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  [. A  /  x ]. ph ) )
5 intss1 3756 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl6bir 163 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  ->  |^| { x  |  ph }  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   {cab 2103   F/_wnfc 2245   [.wsbc 2882    C_ wss 3041   |^|cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by: (None)
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