![]() |
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. Version of 19.3 2200 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1981 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 2202 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1776 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: exlimd 2216 19.19 2227 19.36 2228 19.41 2233 19.44 2235 19.45 2236 19.9h 2285 eeor 2334 dfid3 5586 bnj1189 35002 bj-exexbiex 36683 bj-exalbial 36685 ax6e2ndeq 44557 e2ebind 44561 ax6e2ndeqVD 44907 e2ebindVD 44910 e2ebindALT 44927 ax6e2ndeqALT 44929 |
Copyright terms: Public domain | W3C validator |