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Mirrors > Home > MPE Home > Th. List > 19.9t | Structured version Visualization version GIF version |
Description: Closed form of 19.9 2190 and version of 19.3t 2185 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) |
Ref | Expression |
---|---|
19.9t | ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
2 | 1 | 19.9d 2187 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
3 | 19.8a 2166 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 2, 3 | impbid1 217 | 1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ 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: 19.9 2190 19.21t 2191 sbftv 2245 spimtOLD 2351 sbft 2455 vtoclegft 3482 bj-cbv3tb 33307 bj-spimtv 33314 bj-equsal1t 33392 bj-19.21t 33400 |
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