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Theorem sb56 2227
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 1791. The implication "to the left" is equs4 2092 and does not require any dv condition. Theorem equs45f 2148 replaces the dv condition with a non-freeness hypothesis. (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 nfa1 1956 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
2 ax12v 1910 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
3 sp 1914 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 32 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
52, 4impbid 193 . 2  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
61, 5equsex 2095 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  sb6  2228  sb5  2229  mopick  2334  alexeqg  3200  pm13.196a  36735
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