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Theorem sb10f 2188
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.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb10f.1  |-  F/ 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  |-  F/ x ph
21nfsb 2168 . . 3  |-  F/ x [ y  /  z ] ph
3 sbequ 2090 . . 3  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
42, 3equsex 2011 . 2  |-  ( E. x ( x  =  y  /\  [ x  /  z ] ph ) 
<->  [ y  /  z ] ph )
54bicomi 202 1  |-  ( [ y  /  z ]
ph 
<->  E. x ( x  =  y  /\  [
x  /  z ]
ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1596   F/wnf 1599   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712
This theorem is referenced by: (None)
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