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Theorem nrmo 35889
Description: "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
Hypothesis
Ref Expression
nrmo.1 (𝑥𝐴 → ¬ 𝜑)
Assertion
Ref Expression
nrmo ∃*𝑥𝐴 𝜑

Proof of Theorem nrmo
StepHypRef Expression
1 mofal 35888 . . 3 ∃*𝑥
2 nrmo.1 . . . . . . 7 (𝑥𝐴 → ¬ 𝜑)
32imori 853 . . . . . 6 𝑥𝐴 ∨ ¬ 𝜑)
4 ianor 980 . . . . . 6 (¬ (𝑥𝐴𝜑) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝜑))
53, 4mpbir 230 . . . . 5 ¬ (𝑥𝐴𝜑)
65bifal 1550 . . . 4 ((𝑥𝐴𝜑) ↔ ⊥)
76mobii 2538 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥⊥)
81, 7mpbir 230 . 2 ∃*𝑥(𝑥𝐴𝜑)
9 df-rmo 3372 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
108, 9mpbir 230 1 ∃*𝑥𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846  wfal 1546  wcel 2099  ∃*wmo 2528  ∃*wrmo 3371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-mo 2530  df-rmo 3372
This theorem is referenced by: (None)
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