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Theorem csbeq2gOLD 36959
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3793. csbeq2gOLD 36959 is derived from the virtual deduction proof csbeq2gVD 37328. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3378 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbeq2gOLD  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )

Proof of Theorem csbeq2gOLD
StepHypRef Expression
1 spsbc 3291 . 2  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
2 sbceqg 3784 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
31, 2sylibd 222 1  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452    = wceq 1454    e. wcel 1897   [.wsbc 3278   [_csb 3374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-csb 3375
This theorem is referenced by:  csbsngVD  37329  csbxpgVD  37330  csbresgVD  37331  csbrngVD  37332  csbima12gALTVD  37333  csbfv12gALTVD  37335
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