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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfbi | Structured version Visualization version GIF version |
Description: Closed form of nfbii 1770 (with df-bj-nf 31765 instead of nf5 2102, which would require more axioms). (Contributed by BJ, 6-May-2019.) |
Ref | Expression |
---|---|
bj-nfbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbi 1762 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) | |
2 | albi 1736 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) | |
3 | 1, 2 | imbi12d 333 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓))) |
4 | df-bj-nf 31765 | . 2 ⊢ (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
5 | df-bj-nf 31765 | . 2 ⊢ (ℲℲ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
6 | 3, 4, 5 | 3bitr4g 302 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 ℲℲwnff 31764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-bj-nf 31765 |
This theorem is referenced by: bj-nfxfr 31794 |
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