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Theorem csbeq2d 3842
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 2527 . . . 4  |-  ( ph  ->  ( y  e.  B  <->  y  e.  C ) )
41, 3sbcbid 3385 . . 3  |-  ( ph  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2593 . 2  |-  ( ph  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
6 df-csb 3431 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
7 df-csb 3431 . 2  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
85, 6, 73eqtr4g 2523 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395   F/wnf 1617    e. wcel 1819   {cab 2442   [.wsbc 3327   [_csb 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-sbc 3328  df-csb 3431
This theorem is referenced by:  csbeq2dv  3843
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