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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbntbi | Structured version Visualization version GIF version |
Description: Strengthening hbnt 2302 by replacing its succedent with a biconditional. See also hbntg 33052 and hbntal 40894. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 34039. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-hbntbi | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-19.9htbi 34039 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) | |
2 | 1 | bicomd 225 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑)) |
3 | 2 | notbid 320 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑)) |
4 | alnex 1782 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | 3, 4 | syl6bbr 291 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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