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Theorem sb56 1813
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 1693. (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 1478 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
2 ax11v 1755 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
3 ax-4 1445 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 30 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
52, 4impbid 127 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
61, 5equsex 1663 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sb6  1814  sb5  1815  alexeq  2743  dfdif3  3110
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