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Theorem sbequ2 1728
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
Assertion
Ref Expression
sbequ2  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )

Proof of Theorem sbequ2
StepHypRef Expression
1 df-sb 1727 . . 3  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
21simplbi 460 . 2  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  ph ) )
32com12 31 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1599   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-sb 1727
This theorem is referenced by:  stdpc7  1787  sbequ12  1978  dfsb2  2100  sbequi  2102  sbi1  2119  mo3OLD  2310  mopickOLD  2343  2pm13.193  33193  2pm13.193VD  33571
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