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Theorem sbcssg 3358
Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcssg  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )

Proof of Theorem sbcssg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcalg 2867 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C )  <->  A. y [. A  /  x ]. ( y  e.  B  ->  y  e.  C ) ) )
2 sbcimg 2856 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
) ) )
3 sbcel2g 2928 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
4 sbcel2g 2928 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) )
53, 4imbi12d 232 . . . . 5  |-  ( A  e.  V  ->  (
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
)  <->  ( y  e. 
[_ A  /  x ]_ B  ->  y  e. 
[_ A  /  x ]_ C ) ) )
62, 5bitrd 186 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
76albidv 1746 . . 3  |-  ( A  e.  V  ->  ( A. y [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <->  A. y
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
81, 7bitrd 186 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C )  <->  A. y
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
9 dfss2 2989 . . 3  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
109sbcbii 2874 . 2  |-  ( [. A  /  x ]. B  C_  C  <->  [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C ) )
11 dfss2 2989 . 2  |-  ( [_ A  /  x ]_ B  C_ 
[_ A  /  x ]_ C  <->  A. y ( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) )
128, 10, 113bitr4g 221 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    e. wcel 1434   [.wsbc 2816   [_csb 2909    C_ wss 2974
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910  df-in 2980  df-ss 2987
This theorem is referenced by:  sbcrel  4452  sbcfg  5076
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