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Mirrors > Home > MPE Home > Th. List > 19.21t | Structured version Visualization version GIF version |
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2196. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1779 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.) |
Ref | Expression |
---|---|
19.21t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38a 1835 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | |
2 | 19.9t 2193 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 2 | imbi1d 341 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
4 | 1, 3 | bitr3d 281 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 |
This theorem is referenced by: 19.21 2196 sbal1 2523 sbal2 2524 r19.21t 3247 ceqsal1t 3502 sbciegftOLD 3815 bj-ceqsalt0 36362 bj-ceqsalt1 36363 wl-sbhbt 37021 wl-2sb6d 37025 wl-sbalnae 37029 ax12indalem 38417 ax12inda2ALT 38418 |
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