Theorem nrmo 34578

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	⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑)

Assertion

Ref	Expression
nrmo	⊢ ∃*𝑥 ∈ 𝐴 𝜑

Proof of Theorem nrmo

Step	Hyp	Ref	Expression
1		mofal 34577	. . 3 ⊢ ∃*𝑥⊥
2		nrmo.1	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑)
3	2	imori 850	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑)
4		ianor 978	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑))
5	3, 4	mpbir 230	. . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)
6	5	bifal 1557	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ⊥)
7	6	mobii 2549	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥⊥)
8	1, 7	mpbir 230	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
9		df-rmo 3073	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10	8, 9	mpbir 230	1 ⊢ ∃*𝑥 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843  ⊥wfal 1553   ∈ wcel 2109  ∃*wmo 2539  ∃*wrmo 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-mo 2541  df-rmo 3073

This theorem is referenced by: (None)