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Theorem moanim 2682
 Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2681. (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 . . 3 𝑥𝜑
2 ibar 532 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2mobid 2609 . 2 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
4 simpl 486 . . 3 ((𝜑𝜓) → 𝜑)
51, 4exlimi 2215 . 2 (∃𝑥(𝜑𝜓) → 𝜑)
63, 5moanimlem 2680 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785  ∃*wmo 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2598 This theorem is referenced by:  moanmo  2684  reuxfrdf  30272
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