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Theorem nrmo 33751
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 33750 . . 3 ∃*𝑥
2 nrmo.1 . . . . . . 7 (𝑥𝐴 → ¬ 𝜑)
32imori 850 . . . . . 6 𝑥𝐴 ∨ ¬ 𝜑)
4 ianor 978 . . . . . 6 (¬ (𝑥𝐴𝜑) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝜑))
53, 4mpbir 233 . . . . 5 ¬ (𝑥𝐴𝜑)
65bifal 1547 . . . 4 ((𝑥𝐴𝜑) ↔ ⊥)
76mobii 2625 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥⊥)
81, 7mpbir 233 . 2 ∃*𝑥(𝑥𝐴𝜑)
9 df-rmo 3144 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
108, 9mpbir 233 1 ∃*𝑥𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843  wfal 1543  wcel 2108  ∃*wmo 2614  ∃*wrmo 3139
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 1905  ax-6 1964
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-fal 1544  df-ex 1775  df-mo 2616  df-rmo 3144
This theorem is referenced by: (None)
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