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Theorem mo2v 2614
Description: Alternate definition of "at most one." Unlike mo2 2616, which is slightly more general, it does not depend on ax-11 2183 and ax-13 2391, 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 2612 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-eu 2611 . . 3 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32imbi2i 325 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
4 alnex 1855 . . . . . . 7 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 pm2.21 120 . . . . . . . 8 𝜑 → (𝜑𝑥 = 𝑦))
65alimi 1888 . . . . . . 7 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
74, 6sylbir 225 . . . . . 6 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
87eximi 1911 . . . . 5 (∃𝑦 ¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
9819.23bi 2208 . . . 4 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
10 biimp 205 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
1110alimi 1888 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1211eximi 1911 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
139, 12ja 173 . . 3 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
14 nfia1 2179 . . . . . 6 𝑥(∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
15 id 22 . . . . . . . . . 10 (𝜑𝜑)
16 ax12v 2197 . . . . . . . . . . 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 2204 . . . . . . . 8 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑)))
2019ancld 577 . . . . . . 7 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑))))
21 albiim 1965 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
2220, 21syl6ibr 242 . . . . . 6 (𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2314, 22exlimi 2233 . . . . 5 (∃𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2423eximdv 1995 . . . 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 383  wal 1630  wex 1853  ∃!weu 2607  ∃*wmo 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612
This theorem is referenced by:  mo2  2616  eu3v  2635  mo3  2645  sbmo  2653  moim  2657  mopick  2673  2mo2  2688  mo2icl  3526  moabex  5076  dffun3  6060  dffun6f  6063  grothprim  9868  bj-mo3OLD  33160  wl-mo2df  33683  wl-mo2t  33688  wl-mo3t  33689  dffrege115  38792
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