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 ) and the converse implication is the join of instances of bj-alrimg and bj-exlimg (see 19.38a and 19.38b ). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-nfimexal | ⊢ ( ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ∨ ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.38 | ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) | |
| 2 | bj-alrimg | ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) | |
| 3 | bj-exlimg | ⊢ ( ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) | |
| 4 | 2 3 | jaoi | ⊢ ( ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ∨ ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) |
| 5 | 1 4 | impbid2 | ⊢ ( ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ∨ ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |