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Theorem mo2v 2475
Description: Alternate definition of "at most one." Unlike mo2 2477, which is slightly more general, it does not depend on ax-11 2032 and ax-13 2244, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)
Assertion
Ref Expression
mo2v (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem mo2v
StepHypRef Expression
1 df-mo 2473 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-eu 2472 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32imbi2i 326 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 alnex 1704 . . . . . . 7 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 pm2.21 120 . . . . . . . 8 𝜑 → (𝜑𝑥 = 𝑦))
65alimi 1737 . . . . . . 7 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
74, 6sylbir 225 . . . . . 6 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
87eximi 1760 . . . . 5 (∃𝑦 ¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
9819.23bi 2059 . . . 4 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
10 biimp 205 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
1110alimi 1737 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1211eximi 1760 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
139, 12ja 173 . . 3 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
14 nfia1 2028 . . . . . 6 𝑥(∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
15 id 22 . . . . . . . . . 10 (𝜑𝜑)
16 ax12v 2046 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
1716com12 32 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
1815, 17embantd 59 . . . . . . . . 9 (𝜑 → ((𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
1918spsd 2055 . . . . . . . 8 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
2019ancld 575 . . . . . . 7 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑))))
21 albiim 1814 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
2220, 21syl6ibr 242 . . . . . 6 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2314, 22exlimi 2084 . . . . 5 (∃𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2423eximdv 1844 . . . 4 (∃𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
2524com12 32 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
2613, 25impbii 199 . 2 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
271, 3, 263bitri 286 1 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wex 1702  ∃!weu 2468  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473
This theorem is referenced by:  mo2  2477  eu3v  2496  mo3  2505  sbmo  2513  moim  2517  mopick  2533  2mo2  2548  mo2icl  3379  moabex  4918  dffun3  5887  dffun6f  5890  grothprim  9641  bj-mo3OLD  32807  wl-mo2df  33323  wl-mo2t  33328  wl-mo3t  33329  dffrege115  38092
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