![]() |
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 2196 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1988 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 2198 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 |
This theorem is referenced by: exlimd 2212 19.19 2223 19.36 2224 19.41 2229 19.44 2231 19.45 2232 19.9h 2283 eeor 2330 dfid3 5578 bnj1189 34020 bj-exexbiex 35578 bj-exalbial 35580 ax6e2ndeq 43320 e2ebind 43324 ax6e2ndeqVD 43670 e2ebindVD 43673 e2ebindALT 43690 ax6e2ndeqALT 43692 |
Copyright terms: Public domain | W3C validator |