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Theorem sbcssOLD 33414
Description: Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 33784. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcssOLD  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )

Proof of Theorem sbcssOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfss2 3488 . . . 4  |-  ( C 
C_  D  <->  A. y
( y  e.  C  ->  y  e.  D ) )
21sbcbiiOLD 3388 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  [. A  /  x ]. A. y ( y  e.  C  -> 
y  e.  D ) ) )
3 sbcalgOLD 3380 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D ) ) )
4 sbcimg 3369 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <-> 
( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D
) ) )
5 sbcel2gOLD 3840 . . . . . . . 8  |-  ( A  e.  B  ->  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) )
6 sbcel2gOLD 3840 . . . . . . . 8  |-  ( A  e.  B  ->  ( [. A  /  x ]. y  e.  D  <->  y  e.  [_ A  /  x ]_ D ) )
75, 6imbi12d 320 . . . . . . 7  |-  ( A  e.  B  ->  (
( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D
)  <->  ( y  e. 
[_ A  /  x ]_ C  ->  y  e. 
[_ A  /  x ]_ D ) ) )
84, 7bitrd 253 . . . . . 6  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <-> 
( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) )
98alrimiv 1720 . . . . 5  |-  ( A  e.  B  ->  A. y
( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <-> 
( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) )
10 albi 1640 . . . . 5  |-  ( A. y ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e. 
[_ A  /  x ]_ D ) )  -> 
( A. y [. A  /  x ]. (
y  e.  C  -> 
y  e.  D )  <->  A. y ( y  e. 
[_ A  /  x ]_ C  ->  y  e. 
[_ A  /  x ]_ D ) ) )
119, 10syl 16 . . . 4  |-  ( A  e.  B  ->  ( A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  A. y
( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) )
123, 11bitrd 253 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y
( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) )
132, 12bitrd 253 . 2  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  A. y
( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) )
14 dfss2 3488 . 2  |-  ( [_ A  /  x ]_ C  C_ 
[_ A  /  x ]_ D  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) )
1513, 14syl6bbr 263 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    e. wcel 1819   [.wsbc 3327   [_csb 3430    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-in 3478  df-ss 3485
This theorem is referenced by: (None)
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