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Theorem mobi 2549
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 1896 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
2 moim 2546 . . . 4 (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓))
3 moim 2546 . . . 4 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
42, 3impbid21d 210 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)))
54imp 407 . 2 ((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
61, 5sylbi 216 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540  ∃*wmo 2540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-mo 2542
This theorem is referenced by:  mobidv  2551  mobid  2552  eubi  2586
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