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Theorem moanim 2690
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 520 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2mobid 2648 . . 3 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
43biimprcd 241 . 2 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
5 simpl 470 . . . . . 6 ((𝜑𝜓) → 𝜑)
61, 5exlimi 2252 . . . . 5 (∃𝑥(𝜑𝜓) → 𝜑)
7 exmo 2654 . . . . . 6 (∃𝑥(𝜑𝜓) ∨ ∃*𝑥(𝜑𝜓))
87ori 879 . . . . 5 (¬ ∃𝑥(𝜑𝜓) → ∃*𝑥(𝜑𝜓))
96, 8nsyl4 157 . . . 4 (¬ ∃*𝑥(𝜑𝜓) → 𝜑)
109con1i 146 . . 3 𝜑 → ∃*𝑥(𝜑𝜓))
11 moan 2685 . . 3 (∃*𝑥𝜓 → ∃*𝑥(𝜑𝜓))
1210, 11ja 174 . 2 ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
134, 12impbii 200 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wex 1859  wnf 1863  ∃*wmo 2630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864  df-eu 2633  df-mo 2634
This theorem is referenced by:  moanimv  2692  moanmo  2693
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