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Mirrors > Home > MPE Home > Th. List > 19.38a | Structured version Visualization version GIF version |
Description: Under a non-freeness hypothesis, the implication 19.38 1766 can be strengthened to an equivalence. See also 19.38b 1768. (Contributed by BJ, 3-Nov-2021.) |
Ref | Expression |
---|---|
19.38a | ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38 1766 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
2 | df-nf 1710 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
3 | alim 1738 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | |
4 | imim1 83 | . . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ((∀𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
5 | 3, 4 | syl5 34 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
6 | 2, 5 | sylbi 207 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
7 | 1, 6 | impbid2 216 | 1 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-ex 1705 df-nf 1710 |
This theorem is referenced by: 19.21t 2073 |
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