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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.9htbi | Structured version Visualization version GIF version | ||
| Description: Strengthening 19.9ht 2320 by replacing its consequent with a biconditional (19.9t 2204 does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019.) |
| Ref | Expression |
|---|---|
| bj-19.9htbi | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.9ht 2320 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
| 2 | 19.8a 2181 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
| 3 | 1, 2 | impbid1 225 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| 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-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: bj-hbntbi 36698 |
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