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Theorem sbcssg 3928
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 3374 . . 3  |-  ( [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C )  <->  A. y [. A  /  x ]. ( y  e.  B  ->  y  e.  C ) )
2 sbcimg 3366 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  ->  y  e.  C )  <-> 
( [. A  /  x ]. y  e.  B  ->  [. A  /  x ]. y  e.  C
) ) )
3 sbcel2 3828 . . . . . 6  |-  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B
)
4 sbcel2 3828 . . . . . 6  |-  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C
)
53, 4imbi12i 324 . . . . 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 1718 . . 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 3478 . . 3  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
109sbcbii 3380 . 2  |-  ( [. A  /  x ]. B  C_  C  <->  [. A  /  x ]. A. y ( y  e.  B  ->  y  e.  C ) )
11 dfss2 3478 . 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 1396    e. wcel 1823   [.wsbc 3324   [_csb 3420    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784
This theorem is referenced by:  sbcrel  5077  sbcfg  5711  iuninc  27638  cotrclrcl  38232
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