Theorem rals2d

Description: Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent ps , not merely be a member of A . (Contributed by David A. Wheeler, 20-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypothesis	rals2d.1	⊢ ( 𝜑 → ∀∃ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
Assertion	rals2d	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rals2d.1	⊢ ( 𝜑 → ∀∃ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
2		df-rals	⊢ ( ∀∃ 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 )