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Theorem moanim 2622
Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2621. (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 529 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2mobid 2550 . 2 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
4 simpl 483 . . 3 ((𝜑𝜓) → 𝜑)
51, 4exlimi 2210 . 2 (∃𝑥(𝜑𝜓) → 𝜑)
63, 5moanimlem 2620 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wnf 1786  ∃*wmo 2538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-mo 2540
This theorem is referenced by:  moanmo  2624  reuxfrdf  30839
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