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Mirrors > Home > MPE Home > Th. List > 19.3t | Structured version Visualization version GIF version |
Description: Closed form of 19.3 2193 and version of 19.9t 2195 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 2174 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | nf5r 2185 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | impbid2 225 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 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 1911 ax-6 1969 ax-7 2009 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-ex 1780 df-nf 1784 |
This theorem is referenced by: 19.23t 2201 |
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