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Mirrors > Home > MPE Home > Th. List > 19.3t | Structured version Visualization version GIF version |
Description: Closed form of 19.3 2200 and version of 19.9t 2202 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 2181 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | nf5r 2192 | . 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 1535 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-ex 1777 df-nf 1781 |
This theorem is referenced by: 19.23t 2208 |
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