Theorem sb10f 2494

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 2488	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝜑
3		sbequ 2035	. . 3 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
4	2, 3	equsexv 2197	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑) ↔ [𝑦 / 𝑧]𝜑)
5	4	bicomi 216	1 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))

Syntax hints:   ↔ wb 198   ∧ wa 387  ∃wex 1742  Ⅎwnf 1746  [wsb 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016

This theorem is referenced by: (None)