Theorem rals2d 50454

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 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

Hypothesis

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

Assertion

Ref	Expression
rals2d	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Proof of Theorem rals2d

Step	Hyp	Ref	Expression
1		rals2d.1	. . 3 ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
2		df-rals 50447	. . 3 ⊢ (∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃𝑥 ∈ 𝐴 𝜓))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃𝑥 ∈ 𝐴 𝜓))
4	3	simprd 500	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∀wral 3085  ∃wrex 3095  ∀∃wrals 50445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-rals 50447

This theorem is referenced by:  ralsn0d  50455