Theorem ralseu2d

Description: Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent ps , not merely be a member of A . (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Hypothesis	ralseu2d.1	⊢ ( 𝜑 → ∀∃! 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
Assertion	ralseu2d	⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralseu2d.1	⊢ ( 𝜑 → ∀∃! 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
2		df-ralseu	⊢ ( ∀∃! 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) ∧ ∃! 𝑥 ∈ 𝐴 𝜓 ) )
3	1 2	sylib	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) ∧ ∃! 𝑥 ∈ 𝐴 𝜓 ) )
4	3	simprd	⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜓 )