| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 19.3t | Structured version Visualization version GIF version | ||
| Description: Closed form of 19.3 2202 and version of 19.9t 2204 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.) |
| Ref | Expression |
|---|---|
| 19.3t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp 2183 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
| 2 | nf5r 2194 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
| 3 | 1, 2 | impbid2 226 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 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: 19.23t 2210 |
| Copyright terms: Public domain | W3C validator |