![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 19.9t | Structured version Visualization version GIF version |
Description: A closed version of 19.9 2228. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) |
Ref | Expression |
---|---|
19.9t | ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
2 | 1 | 19.9d 2225 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
3 | 19.8a 2206 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 2, 3 | impbid1 215 | 1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∃wex 1852 Ⅎwnf 1856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-ex 1853 df-nf 1858 |
This theorem is referenced by: 19.9 2228 19.21t 2229 19.21tOLDOLD 2230 spimt 2415 sbft 2526 vtoclegft 3431 bj-cbv3tb 33048 bj-spimtv 33055 bj-sbftv 33099 bj-equsal1t 33144 bj-19.21t 33152 19.9alt 34774 |
Copyright terms: Public domain | W3C validator |