Theorem ralsd

Description: Introduction rule for "all some" restricted to a class. This is the converse of rals1d and rals2d taken together. (Contributed by David A. Wheeler, 12-Jul-2026)

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

Proof

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