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Theorem csbeq2g 28710
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3279. csbeq2g 28710 is derived from the virtual deduction proof csbeq2gVD 29078. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbeq2g  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )

Proof of Theorem csbeq2g
StepHypRef Expression
1 spsbc 3175 . 2  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
2 sbceqg 3269 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
31, 2sylibd 207 1  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  csbsngVD  29079  csbxpgVD  29080  csbresgVD  29081  csbrngVD  29082  csbima12gALTVD  29083  csbfv12gALTVD  29085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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