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Theorem moanim 2517
 Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
Hypothesis
Ref Expression
moanim.1 𝑥𝜑
Assertion
Ref Expression
moanim (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Proof of Theorem moanim
StepHypRef Expression
1 moanim.1 . . . 4 𝑥𝜑
2 ibar 524 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2mobid 2477 . . 3 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
43biimprcd 239 . 2 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
5 simpl 472 . . . . . 6 ((𝜑𝜓) → 𝜑)
61, 5exlimi 2073 . . . . 5 (∃𝑥(𝜑𝜓) → 𝜑)
7 exmo 2483 . . . . . 6 (∃𝑥(𝜑𝜓) ∨ ∃*𝑥(𝜑𝜓))
87ori 389 . . . . 5 (¬ ∃𝑥(𝜑𝜓) → ∃*𝑥(𝜑𝜓))
96, 8nsyl4 155 . . . 4 (¬ ∃*𝑥(𝜑𝜓) → 𝜑)
109con1i 143 . . 3 𝜑 → ∃*𝑥(𝜑𝜓))
11 moan 2512 . . 3 (∃*𝑥𝜓 → ∃*𝑥(𝜑𝜓))
1210, 11ja 172 . 2 ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
134, 12impbii 198 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699  ∃*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:  moanimv  2519  moanmo  2520
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