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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 2186 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2030 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 2189 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∃wex 1823 Ⅎwnf 1827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-ex 1824 df-nf 1828 |
This theorem is referenced by: exlimd 2204 19.19 2215 19.36 2216 19.41 2221 19.44 2224 19.45 2225 19.9h 2260 euaeOLD 2691 dfid3 5264 fsplit 7565 bnj1189 31680 bj-exexbiex 33283 bj-exalbial 33285 ax6e2ndeq 39729 e2ebind 39733 ax6e2ndeqVD 40088 e2ebindVD 40091 e2ebindALT 40108 ax6e2ndeqALT 40110 |
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