ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbequ2 Unicode version

Theorem sbequ2 1769
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ2  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )

Proof of Theorem sbequ2
StepHypRef Expression
1 df-sb 1763 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
2 simpl 109 . . 3  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ( x  =  y  ->  ph )
)
32com12 30 . 2  |-  ( x  =  y  ->  (
( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ph ) )
41, 3biimtrid 152 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1492   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  stdpc7  1770  sbequ12  1771  sbequi  1839  mo23  2067  mopick  2104
  Copyright terms: Public domain W3C validator