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| 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 2202 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1983 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 2204 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1779 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 1910 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: exlimd 2218 19.19 2229 19.36 2230 19.41 2235 19.44 2237 19.45 2238 19.9h 2286 eeor 2335 dfid3 5581 bnj1189 35023 bj-exexbiex 36701 bj-exalbial 36703 ax6e2ndeq 44579 e2ebind 44583 ax6e2ndeqVD 44929 e2ebindVD 44932 e2ebindALT 44949 ax6e2ndeqALT 44951 | 
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