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Theorem bm1.1 2636
Description: Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
Hypothesis
Ref Expression
bm1.1.1 𝑥𝜑
Assertion
Ref Expression
bm1.1 (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bm1.1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 biantr 992 . . . . 5 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
21alanimi 1784 . . . 4 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
3 ax-ext 2631 . . . 4 (∀𝑦(𝑦𝑥𝑦𝑧) → 𝑥 = 𝑧)
42, 3syl 17 . . 3 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
54gen2 1763 . 2 𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
6 nfv 1883 . . . . . 6 𝑥 𝑦𝑧
7 bm1.1.1 . . . . . 6 𝑥𝜑
86, 7nfbi 1873 . . . . 5 𝑥(𝑦𝑧𝜑)
98nfal 2191 . . . 4 𝑥𝑦(𝑦𝑧𝜑)
10 elequ2 2044 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1110bibi1d 332 . . . . 5 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
1211albidv 1889 . . . 4 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
139, 12mo4f 2545 . . 3 (∃*𝑥𝑦(𝑦𝑥𝜑) ↔ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧))
14 df-mo 2503 . . 3 (∃*𝑥𝑦(𝑦𝑥𝜑) ↔ (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑)))
1513, 14bitr3i 266 . 2 (∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧) ↔ (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑)))
165, 15mpbi 220 1 (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744  wnf 1748  ∃!weu 2498  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503
This theorem is referenced by:  zfnuleu  4819
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