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Theorem sbequ2 1809
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 1808 . . 3  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
21simplbi 466 . 2  |-  ( [ y  /  x ] ph  ->  ( x  =  y  ->  ph ) )
32com12 32 1  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   E.wex 1673   [wsb 1807
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 190  df-an 377  df-sb 1808
This theorem is referenced by:  stdpc7  1881  sbequ12  2093  dfsb2  2212  sbequi  2214  sbi1  2231  bj-mo3OLD  31491  2pm13.193  36962  2pm13.193VD  37339
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