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Theorem sbcssg 3442
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 2933 . . 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 2922 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
) ) )
3 sbcel2g 2994 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
4 sbcel2g 2994 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) )
53, 4imbi12d 233 . . . . 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 187 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
76albidv 1780 . . 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 187 . 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 3056 . . 3  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
109sbcbii 2940 . 2  |-  ( [. A  /  x ]. B  C_  C  <->  [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C ) )
11 dfss2 3056 . 2  |-  ( [_ A  /  x ]_ B  C_ 
[_ A  /  x ]_ C  <->  A. y ( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) )
128, 10, 113bitr4g 222 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 104   A.wal 1314    e. wcel 1465   [.wsbc 2882   [_csb 2975    C_ wss 3041
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-csb 2976  df-in 3047  df-ss 3054
This theorem is referenced by:  sbcrel  4595  sbcfg  5241
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