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Mirrors > Home > ILE Home > Th. List > 19.9ht | GIF version |
Description: A closed version of one direction of 19.9 1591. (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 1393 | . 2 ⊢ ∀𝑥(𝜑 → 𝜑) |
3 | 19.23ht 1441 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))) | |
4 | 2, 3 | mpbii 147 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1297 ∃wex 1436 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-gen 1393 ax-ie2 1438 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.9t 1589 19.9h 1590 19.9hd 1608 |
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