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Theorem csbeq2d 3275
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
csbeq2d.1  |-  F/ x ph
csbeq2d.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
csbeq2d  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )

Proof of Theorem csbeq2d
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq2d.1 . . . 4  |-  F/ x ph
2 csbeq2d.2 . . . . 5  |-  ( ph  ->  B  =  C )
32eleq2d 2503 . . . 4  |-  ( ph  ->  ( y  e.  B  <->  y  e.  C ) )
41, 3sbcbid 3214 . . 3  |-  ( ph  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2550 . 2  |-  ( ph  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
6 df-csb 3252 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
7 df-csb 3252 . 2  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
85, 6, 73eqtr4g 2493 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1553    = wceq 1652    e. wcel 1725   {cab 2422   [.wsbc 3161   [_csb 3251
This theorem is referenced by:  csbeq2dv  3276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-sbc 3162  df-csb 3252
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