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Theorem sbcssg 3891
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 sbcal 3339 . . 3  |-  ( [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C )  <->  A. y [. A  /  x ]. ( y  e.  B  ->  y  e.  C ) )
2 sbcimg 3329 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
) ) )
3 sbcel2 3784 . . . . . 6  |-  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B
)
4 sbcel2 3784 . . . . . 6  |-  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C
)
53, 4imbi12i 326 . . . . 5  |-  ( (
[. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
)  <->  ( y  e. 
[_ A  /  x ]_ B  ->  y  e. 
[_ A  /  x ]_ C ) )
62, 5syl6bb 261 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) ) )
76albidv 1680 . . 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, 7syl5bb 257 . 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 3446 . . 3  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
109sbcbii 3347 . 2  |-  ( [. A  /  x ]. B  C_  C  <->  [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C ) )
11 dfss2 3446 . 2  |-  ( [_ A  /  x ]_ B  C_ 
[_ A  /  x ]_ C  <->  A. y ( y  e.  [_ A  /  x ]_ B  ->  y  e.  [_ A  /  x ]_ C ) )
128, 10, 113bitr4g 288 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    e. wcel 1758   [.wsbc 3287   [_csb 3389    C_ wss 3429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739
This theorem is referenced by:  sbcrel  5027  sbcfg  5658  iuninc  26055
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