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Mirrors > Home > MPE Home > Th. List > 19.9t | Structured version Visualization version GIF version |
Description: A closed version of 19.9 2110. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (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 2108 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
3 | 19.8a 2090 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 2, 3 | impbid1 215 | 1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∃wex 1744 Ⅎwnf 1748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-ex 1745 df-nf 1750 |
This theorem is referenced by: 19.9 2110 19.21t 2111 19.21tOLDOLD 2112 spimt 2289 sbft 2407 vtoclegft 3311 bj-cbv3tb 32836 bj-spimtv 32843 bj-sbftv 32888 bj-equsal1t 32934 bj-19.21t 32942 19.9alt 34570 |
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