Theorem bj-nfimexal 36627

Description: A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1839) and the converse implication is the join of instances of bj-alrimg 36620 and bj-exlimg 36624 (see 19.38a 1840 and 19.38b 1841). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)

Assertion

Ref	Expression
bj-nfimexal	⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem bj-nfimexal

Step	Hyp	Ref	Expression
1		19.38 1839	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
2		bj-alrimg 36620	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
3		bj-exlimg 36624	. . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4	2, 3	jaoi 858	. 2 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5	1, 4	impbid2 226	1 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780

This theorem is referenced by: (None)