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Mirrors > Home > MPE Home > Th. List > sb10f | Structured version Visualization version GIF version |
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 2390. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb10f.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sb10f | ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb10f.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfsb 2565 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝜑 |
3 | sbequ 2090 | . . 3 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | |
4 | 2, 3 | equsexv 2269 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑) ↔ [𝑦 / 𝑧]𝜑) |
5 | 4 | bicomi 226 | 1 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1780 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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