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Theorem moanim 2558
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 2517 . . 3 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
43biimprcd 240 . 2 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
5 simpl 472 . . . . . 6 ((𝜑𝜓) → 𝜑)
61, 5exlimi 2124 . . . . 5 (∃𝑥(𝜑𝜓) → 𝜑)
7 exmo 2523 . . . . . 6 (∃𝑥(𝜑𝜓) ∨ ∃*𝑥(𝜑𝜓))
87ori 389 . . . . 5 (¬ ∃𝑥(𝜑𝜓) → ∃*𝑥(𝜑𝜓))
96, 8nsyl4 156 . . . 4 (¬ ∃*𝑥(𝜑𝜓) → 𝜑)
109con1i 144 . . 3 𝜑 → ∃*𝑥(𝜑𝜓))
11 moan 2553 . . 3 (∃*𝑥𝜓 → ∃*𝑥(𝜑𝜓))
1210, 11ja 173 . 2 ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
134, 12impbii 199 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1744  wnf 1748  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  moanimv  2560  moanmo  2561
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