MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb10f Unicode version

Theorem sb10f 2158
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 2144 . . 3  |-  F/ x [ y  /  z ] ph
3 sbequ 2095 . . 3  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
42, 3equsex 1962 . 2  |-  ( E. x ( x  =  y  /\  [ x  /  z ] ph ) 
<->  [ y  /  z ] ph )
54bicomi 194 1  |-  ( [ y  /  z ]
ph 
<->  E. x ( x  =  y  /\  [
x  /  z ]
ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547   F/wnf 1550   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
  Copyright terms: Public domain W3C validator