![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 19.9 | Structured version Visualization version GIF version |
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2065 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) |
Ref | Expression |
---|---|
19.9.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.9 | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 19.9t 2227 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ 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: exlimd 2243 19.19 2253 19.36 2254 19.41 2259 19.44 2262 19.45 2263 19.9h 2283 exists1 2710 dfid3 5159 fsplit 7434 bnj1189 31416 bj-exexbiex 33029 bj-exalbial 33031 ax6e2ndeq 39301 e2ebind 39305 ax6e2ndeqVD 39668 e2ebindVD 39671 e2ebindALT 39688 ax6e2ndeqALT 39690 |
Copyright terms: Public domain | W3C validator |