Theorem sb10f 2567

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. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)

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 2561	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝜑
3		sbequ 2086	. . 3 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
4	2, 3	equsexv 2265	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑) ↔ [𝑦 / 𝑧]𝜑)
5	4	bicomi 226	1 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1776  Ⅎwnf 1780  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by: (None)