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Theorem bj-intabssel 13049
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
StepHypRef Expression
1 bj-intabssel.nf . . 3  |-  F/_ x A
21nfsbc1 2926 . . 3  |-  F/ x [. A  /  x ]. ph
3 sbceq1a 2918 . . 3  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
41, 2, 3elabgf 2826 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  [. A  /  x ]. ph ) )
5 intss1 3786 . 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 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)
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