Theorem ralseud

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d and ralseu2d taken together. (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Hypotheses	ralseud.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
	ralseud.2	⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜓 )
Assertion	ralseud	⊢ ( 𝜑 → ∀∃! 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )

Proof

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