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| Mirrors > Home > MPE Home > Th. List > 19.9t | Structured version Visualization version GIF version | ||
| Description: Closed form of 19.9 2205 and version of 19.3t 2201 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 2203 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
| 3 | 19.8a 2181 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
| 4 | 2, 3 | impbid1 225 | 1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ 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: 19.9 2205 19.21t 2206 sbft 2270 vtoclegftOLD 3589 bj-cbv3tb 36788 bj-spimtv 36795 bj-equsal1t 36823 bj-19.21t0 36831 19.9dev 42253 |
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