Theorem bj-nfimexal 37259

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 1868) and the converse implication is the join of instances of bj-alrimg 37234 and bj-exlimg 37256 (see 19.38a 1869 and 19.38b 1870). 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 1868	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
2		bj-alrimg 37234	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
3		bj-exlimg 37256	. . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4	2, 3	jaoi 870	. 2 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5	1, 4	impbid2 229	1 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 860  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809

This theorem is used by: (None)