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	⊢ Ⅎxφ

Assertion

Ref	Expression
sb10f	⊢ ([y / z]φ ↔ ∃x(x = y ∧ [x / z]φ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sb10f

Step	Hyp	Ref	Expression
1		sb10f.1	. . . 4 ⊢ Ⅎxφ
2	1	nfsb 2109	. . 3 ⊢ Ⅎx[y / z]φ
3		sbequ 2060	. . 3 ⊢ (x = y → ([x / z]φ ↔ [y / z]φ))
4	2, 3	equsex 1962	. 2 ⊢ (∃x(x = y ∧ [x / z]φ) ↔ [y / z]φ)
5	4	bicomi 193	1 ⊢ ([y / z]φ ↔ ∃x(x = y ∧ [x / z]φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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)