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Mirrors > Home > MPE Home > Th. List > 19.38b | Structured version Visualization version GIF version |
Description: Under a nonfreeness hypothesis, the implication 19.38 1845 can be strengthened to an equivalence. See also 19.38a 1846. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.) |
Ref | Expression |
---|---|
19.38b | ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38 1845 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
2 | exim 1840 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | |
3 | id 22 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
4 | 3 | nfrd 1798 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
5 | 2, 4 | syl9r 78 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
6 | 1, 5 | impbid2 225 | 1 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 206 df-ex 1787 df-nf 1791 |
This theorem is referenced by: 19.23t 2207 |
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