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Theorem dfmoeu 2534
Description: An elementary proof of moeu 2581 in disguise, connecting an expression characterizing uniqueness (df-mo 2538) to that of existential uniqueness (eu6 2572). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2535. (Contributed by Wolf Lammen, 27-May-2019.)
Assertion
Ref Expression
dfmoeu ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfmoeu
StepHypRef Expression
1 alnex 1778 . . . . 5 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . . . . 6 𝜑 → (𝜑𝑥 = 𝑦))
32alimi 1808 . . . . 5 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
41, 3sylbir 235 . . . 4 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
5419.8ad 2180 . . 3 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 biimp 215 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
76alimi 1808 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
87eximi 1832 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
95, 8ja 186 . 2 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
10 nfia1 2151 . . . . 5 𝑥(∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
11 id 22 . . . . . . . . 9 (𝜑𝜑)
12 ax12v 2176 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
1312com12 32 . . . . . . . . 9 (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
1411, 13embantd 59 . . . . . . . 8 (𝜑 → ((𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
1514spsd 2185 . . . . . . 7 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
1615ancld 550 . . . . . 6 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑))))
17 albiim 1887 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
1816, 17imbitrrdi 252 . . . . 5 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
1910, 18exlimi 2215 . . . 4 (∃𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2019eximdv 1915 . . 3 (∃𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
2120com12 32 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
229, 21impbii 209 1 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535  wex 1776
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 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  dfeumo  2535  eu6  2572  dfmo  2594
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