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Theorem sbcssg 3330
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 2811 . . 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 2804 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
) ) )
3 sbcel2g 2871 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
4 sbcel2g 2871 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) )
53, 4imbi12d 223 . . . . 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 177 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
76albidv 1705 . . 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 177 . 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 2934 . . 3  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
109sbcbii 2818 . 2  |-  ( [. A  /  x ]. B  C_  C  <->  [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C ) )
11 dfss2 2934 . 2  |-  ( [_ A  /  x ]_ B  C_ 
[_ A  /  x ]_ C  <->  A. y ( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) )
128, 10, 113bitr4g 212 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 98   A.wal 1241    e. wcel 1393   [.wsbc 2764   [_csb 2852    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-in 2924  df-ss 2931
This theorem is referenced by:  sbcrel  4426  sbcfg  5045
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