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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 2207. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1784 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 1840 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | |
| 2 | 19.9t 2204 | . . 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 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: 19.21 2207 sbal1 2533 sbal2 2534 r19.21t 3253 ceqsal1t 3514 sbciegftOLD 3826 bj-ceqsalt0 36885 bj-ceqsalt1 36886 wl-sbhbt 37555 wl-2sb6d 37559 wl-sbalnae 37563 ax12indalem 38946 ax12inda2ALT 38947 | 
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