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Theorem sb56 1885
Description: Two equivalent ways of expressing the proper substitution of 
y for  x in  ph, when  x and  y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1763. (Contributed by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb56  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb56
StepHypRef Expression
1 hba1 1540 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
2 ax11v 1827 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
3 ax-4 1510 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 30 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
52, 4impbid 129 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
61, 5equsex 1728 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   E.wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  sb6  1886  sb5  1887  alexeq  2863  dfdif3  3245
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