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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbxfrbi | Structured version Visualization version GIF version |
Description: Closed form of hbxfrbi 1742. Notes: it is less important than bj-nfbi 31793; it requires sp 2041 (unlike bj-nfbi 31793); there is an obvious version with (∃𝑥𝜑 → 𝜑) instead. (Contributed by BJ, 6-May-2019.) |
Ref | Expression |
---|---|
bj-hbxfrbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2041 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | albi 1736 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) | |
3 | 1, 2 | imbi12d 333 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
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-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: (None) |
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