Theorem rexrals 50536

Description: If a member of 𝐴 satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 50542. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)

Assertion

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

Proof of Theorem rexrals

Step	Hyp	Ref	Expression
1		df-rals 50516	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
2		iba 536	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑)))
3	2	bicomd 226	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
4	1, 3	bitrid 286	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wral 3086  ∃wrex 3096  ∀∃wrals 50514

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 50516

This theorem is referenced by: (None)