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Theorem sb10f 2122
 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 xφ
Assertion
Ref Expression
sb10f ([y / z]φx(x = y [x / z]φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sb10f
StepHypRef Expression
1 sb10f.1 . . . 4 xφ
21nfsb 2109 . . 3 x[y / z]φ
3 sbequ 2060 . . 3 (x = y → ([x / z]φ ↔ [y / z]φ))
42, 3equsex 1962 . 2 (x(x = y [x / z]φ) ↔ [y / z]φ)
54bicomi 193 1 ([y / z]φx(x = y [x / z]φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by: (None)
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