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Theorem mobi 2629
 Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
Assertion
Ref Expression
mobi (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Proof of Theorem mobi
StepHypRef Expression
1 albiim 1890 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
2 moim 2626 . . . 4 (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓))
3 moim 2626 . . . 4 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
42, 3impbid21d 214 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)))
54imp 410 . 2 ((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
61, 5sylbi 220 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃*wmo 2620 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2622 This theorem is referenced by:  mobiiOLD  2631  mobidv  2632  mobid  2633  eubi  2668
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