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Theorem sb10f 1868
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 (φxφ)
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 (φxφ)
21hbsb 1820 . . 3 ([y / z]φx[y / z]φ)
3 sbequ 1718 . . 3 (x = y → ([x / z]φ ↔ [y / z]φ))
42, 3equsex 1613 . 2 (x(x = y [x / z]φ) ↔ [y / z]φ)
54bicomi 123 1 ([y / z]φx(x = y [x / z]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by: (None)
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