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Theorem axextmo 2797
 Description: There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)
Hypothesis
Ref Expression
axextmo.1 𝑥𝜑
Assertion
Ref Expression
axextmo ∃*𝑥𝑦(𝑦𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axextmo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 biantr 805 . . . . 5 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
21alanimi 1818 . . . 4 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
3 ax-ext 2793 . . . 4 (∀𝑦(𝑦𝑥𝑦𝑧) → 𝑥 = 𝑧)
42, 3syl 17 . . 3 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
54gen2 1798 . 2 𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
6 nfv 1916 . . . . 5 𝑥 𝑦𝑧
7 axextmo.1 . . . . 5 𝑥𝜑
86, 7nfbi 1905 . . . 4 𝑥(𝑦𝑧𝜑)
98nfal 2343 . . 3 𝑥𝑦(𝑦𝑧𝜑)
10 elequ2 2130 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1110bibi1d 347 . . . 4 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
1211albidv 1922 . . 3 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
139, 12mo4f 2651 . 2 (∃*𝑥𝑦(𝑦𝑥𝜑) ↔ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧))
145, 13mpbir 234 1 ∃*𝑥𝑦(𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785  ∃*wmo 2621 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623 This theorem is referenced by:  nulmo  2798
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