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Theorem csbeq2g 28315
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3107. csbeq2g 28315 is derived from the virtual deduction proof csbeq2gVD 28668. (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 3003 . 2  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
2 sbceqg 3097 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
31, 2sylibd 205 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 1527    = wceq 1623    e. wcel 1684   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbsngVD  28669  csbxpgVD  28670  csbresgVD  28671  csbrngVD  28672  csbima12gALTVD  28673  csbfv12gALTVD  28675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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