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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfimexal | Structured version Visualization version GIF version |
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 1835) and the converse implication is the join of instances of bj-alrimg 36601 and bj-exlimg 36605 (see 19.38a 1836 and 19.38b 1837). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-nfimexal | ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38 1835 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
2 | bj-alrimg 36601 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
3 | bj-exlimg 36605 | . . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
4 | 2, 3 | jaoi 857 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 ∀wal 1534 ∃wex 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1776 |
This theorem is referenced by: (None) |
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