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Theorem sb10f 2531
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 2371. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb10f.1 𝑥𝜑
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
StepHypRef Expression
1 sb10f.1 . . . 4 𝑥𝜑
21nfsb 2526 . . 3 𝑥[𝑦 / 𝑧]𝜑
3 sbequ 2089 . . 3 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
42, 3equsexv 2265 . 2 (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑) ↔ [𝑦 / 𝑧]𝜑)
54bicomi 227 1 ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1787  wnf 1791  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071
This theorem is referenced by: (None)
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