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Mirrors > Home > ILE Home > Th. List > 19.9ht | GIF version |
Description: A closed version of one direction of 19.9 1532. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.9ht | ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ (φ → φ) | |
2 | 1 | ax-gen 1335 | . 2 ⊢ ∀x(φ → φ) |
3 | 19.23ht 1383 | . 2 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → φ) ↔ (∃xφ → φ))) | |
4 | 2, 3 | mpbii 136 | 1 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-gen 1335 ax-ie2 1380 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 19.9t 1530 19.9h 1531 19.9hd 1549 |
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