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Theorem bj-sbceqgALT 32756
Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3788. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3788, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sbceqgALT  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )

Proof of Theorem bj-sbceqgALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2447 . . . . . 6  |-  ( B  =  C  <->  A. y
( y  e.  B  <->  y  e.  C ) )
21sbcth 3309 . . . . 5  |-  ( A  e.  V  ->  [. A  /  x ]. ( B  =  C  <->  A. y
( y  e.  B  <->  y  e.  C ) ) )
3 sbcbig 3341 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( B  =  C  <->  A. y ( y  e.  B  <->  y  e.  C
) )  <->  ( [. A  /  x ]. B  =  C  <->  [. A  /  x ]. A. y ( y  e.  B  <->  y  e.  C ) ) ) )
42, 3mpbid 210 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [. A  /  x ]. A. y ( y  e.  B  <->  y  e.  C
) ) )
5 sbcal 3346 . . . 4  |-  ( [. A  /  x ]. A. y ( y  e.  B  <->  y  e.  C
)  <->  A. y [. A  /  x ]. ( y  e.  B  <->  y  e.  C ) )
64, 5syl6bb 261 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  A. y [. A  /  x ]. ( y  e.  B  <->  y  e.  C
) ) )
7 sbcbig 3341 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  B  <->  y  e.  C )  <->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C
) ) )
87albidv 1680 . . 3  |-  ( A  e.  V  ->  ( A. y [. A  /  x ]. ( y  e.  B  <->  y  e.  C
)  <->  A. y ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C
) ) )
9 sbcel2 3794 . . . . . 6  |-  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B
)
109a1i 11 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
11 sbcel2 3794 . . . . . 6  |-  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C
)
1211a1i 11 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) )
1310, 12bibi12d 321 . . . 4  |-  ( A  e.  V  ->  (
( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )  <->  ( y  e.  [_ A  /  x ]_ B  <->  y  e.  [_ A  /  x ]_ C ) ) )
1413albidv 1680 . . 3  |-  ( A  e.  V  ->  ( A. y ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C
)  <->  A. y ( y  e.  [_ A  /  x ]_ B  <->  y  e.  [_ A  /  x ]_ C ) ) )
156, 8, 143bitrd 279 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  A. y ( y  e. 
[_ A  /  x ]_ B  <->  y  e.  [_ A  /  x ]_ C
) ) )
16 dfcleq 2447 . 2  |-  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  A. y ( y  e.  [_ A  /  x ]_ B  <->  y  e.  [_ A  /  x ]_ C ) )
1715, 16syl6bbr 263 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   [.wsbc 3294   [_csb 3398
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 1955  ax-ext 2432
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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749
This theorem is referenced by: (None)
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