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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 1841) and the converse implication is the join of instances of bj-alrimg 35588 and bj-exlimg 35592 (see 19.38a 1842 and 19.38b 1843). 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 1841 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
2 | bj-alrimg 35588 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
3 | bj-exlimg 35592 | . . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
4 | 2, 3 | jaoi 855 | . 2 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
5 | 1, 4 | impbid2 225 | 1 ⊢ (((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 ∀wal 1539 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-or 846 df-ex 1782 |
This theorem is referenced by: (None) |
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