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Theorem sb10f 2007
Description: Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.)
Hypothesis
Ref Expression
sb10f.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
sb10f  |-  ( [ y  /  z ]
ph 
<->  E. x ( x  =  y  /\  [
x  /  z ]
ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sb10f
StepHypRef Expression
1 sb10f.1 . . . 4  |-  ( ph  ->  A. x ph )
21hbsb 1961 . . 3  |-  ( [ y  /  z ]
ph  ->  A. x [ y  /  z ] ph )
3 sbequ 1851 . . 3  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
42, 3equsex 1739 . 2  |-  ( E. x ( x  =  y  /\  [ x  /  z ] ph ) 
<->  [ y  /  z ] ph )
54bicomi 132 1  |-  ( [ y  /  z ]
ph 
<->  E. x ( x  =  y  /\  [
x  /  z ]
ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362   E.wex 1503   [wsb 1773
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by: (None)
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