Theorem ralsex 50457

Description: The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 50456. (Contributed by David A. Wheeler, 12-Jul-2026.)

Assertion

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

Proof of Theorem ralsex

Step	Hyp	Ref	Expression
1		df-rals 50447	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
2		rexim 3112	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
3	2	imp 411	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜓)
4	1, 3	sylbi 220	1 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃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  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-ral 3086  df-rex 3096  df-rals 50447

This theorem is referenced by: (None)