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Theorem mo2v 2465
 Description: Alternate definition of "at most one." Unlike mo2 2467, which is slightly more general, it does not depend on ax-11 2021 and ax-13 2234, 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 2463 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-eu 2462 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32imbi2i 325 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 alnex 1697 . . . . . . 7 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 pm2.21 119 . . . . . . . 8 𝜑 → (𝜑𝑥 = 𝑦))
65alimi 1730 . . . . . . 7 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
74, 6sylbir 224 . . . . . 6 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
87eximi 1752 . . . . 5 (∃𝑦 ¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
9819.23bi 2049 . . . 4 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
10 biimp 204 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
1110alimi 1730 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1211eximi 1752 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
139, 12ja 172 . . 3 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
14 nfia1 2017 . . . . . 6 𝑥(∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
15 id 22 . . . . . . . . . 10 (𝜑𝜑)
16 ax12v 2035 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
1716com12 32 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
1815, 17embantd 57 . . . . . . . . 9 (𝜑 → ((𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
1918spsd 2045 . . . . . . . 8 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
2019ancld 574 . . . . . . 7 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑))))
21 albiim 1806 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
2220, 21syl6ibr 241 . . . . . 6 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2314, 22exlimi 2073 . . . . 5 (∃𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2423eximdv 1833 . . . 4 (∃𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
2524com12 32 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
2613, 25impbii 198 . 2 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
271, 3, 263bitri 285 1 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  ∃!weu 2458  ∃*wmo 2459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by:  mo2  2467  eu3v  2486  mo3  2495  sbmo  2503  moim  2507  mopick  2523  2mo2  2538  mo2icl  3352  moabex  4854  dffun3  5815  dffun6f  5818  grothprim  9535  bj-mo3OLD  32022  wl-mo2df  32531  wl-mo2t  32536  wl-mo3t  32537  dffrege115  37292
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