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Theorem bj-elssuniab 15228
Description: Version of elssuni 3863 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
Hypothesis
Ref Expression
bj-elssuniab.nf  |-  F/_ x A
Assertion
Ref Expression
bj-elssuniab  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  ->  A  C_  U. {
x  |  ph }
) )

Proof of Theorem bj-elssuniab
StepHypRef Expression
1 sbc8g 2993 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
2 elssuni 3863 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  C_ 
U. { x  | 
ph } )
31, 2biimtrdi 163 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  ->  A  C_  U. {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164   {cab 2179   F/_wnfc 2323   [.wsbc 2985    C_ wss 3153   U.cuni 3835
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-in 3159  df-ss 3166  df-uni 3836
This theorem is referenced by: (None)
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