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Theorem csbeq2g 28347
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3237. csbeq2g 28347 is derived from the virtual deduction proof csbeq2gVD 28713. (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 3133 . 2  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
2 sbceqg 3227 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
31, 2sylibd 206 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 1546    = wceq 1649    e. wcel 1721   [.wsbc 3121   [_csb 3211
This theorem is referenced by:  csbsngVD  28714  csbxpgVD  28715  csbresgVD  28716  csbrngVD  28717  csbima12gALTVD  28718  csbfv12gALTVD  28720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122  df-csb 3212
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