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Mirrors > Home > ILE Home > Th. List > 19.9t | GIF version |
Description: A closed version of 19.9 1644. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
19.9t | ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1461 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | 19.9ht 1641 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
3 | 1, 2 | sylbi 121 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
4 | 19.8a 1590 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
5 | 3, 4 | impbid1 142 | 1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 Ⅎwnf 1460 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: 19.9d 1661 19.23t 1677 spimt 1736 exdistrfor 1800 sbequi 1839 sbft 1848 vtoclegft 2809 copsexg 4243 |
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