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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbntbi | Structured version Visualization version GIF version |
Description: Strengthening hbnt 2129 by replacing its succedent with a biconditional. See also hbntg 30955 and hbntal 37790. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 31881. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-hbntbi | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-19.9htbi 31881 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) | |
2 | 1 | bicomd 212 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 ↔ ∃𝑥𝜑)) |
3 | 2 | notbid 307 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ¬ ∃𝑥𝜑)) |
4 | alnex 1697 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | 3, 4 | syl6bbr 277 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: (None) |
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