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Theorem csbeq2g 28062
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3193. csbeq2g 28062 is derived from the virtual deduction proof csbeq2gVD 28432. (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 3089 . 2  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
2 sbceqg 3183 . 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 1545    = wceq 1647    e. wcel 1715   [.wsbc 3077   [_csb 3167
This theorem is referenced by:  csbsngVD  28433  csbxpgVD  28434  csbresgVD  28435  csbrngVD  28436  csbima12gALTVD  28437  csbfv12gALTVD  28439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-sbc 3078  df-csb 3168
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