ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbeq2d Unicode version

Theorem csbeq2d 2995
Description: Formula-building deduction 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 2185 . . . 4  |-  ( ph  ->  ( y  e.  B  <->  y  e.  C ) )
41, 3sbcbid 2936 . . 3  |-  ( ph  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2233 . 2  |-  ( ph  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
6 df-csb 2974 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
7 df-csb 2974 . 2  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
85, 6, 73eqtr4g 2173 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   F/wnf 1419    e. wcel 1463   {cab 2101   [.wsbc 2880   [_csb 2973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-sbc 2881  df-csb 2974
This theorem is referenced by:  csbeq2dv  2996
  Copyright terms: Public domain W3C validator