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Mirrors > Home > MPE Home > Th. List > nf5 | Structured version Visualization version GIF version |
Description: Alternate definition of df-nf 1785. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1785 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
nf5 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1785 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfa1 2155 | . . 3 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
3 | 2 | 19.23 2211 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 |
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-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: drnf1v 2389 drnf1 2465 axie2 2788 xfree 30223 bj-nfdt0 34031 bj-nfalt 34047 bj-nfext 34048 bj-nfs1t 34114 bj-sbnf 34166 wl-sbnf1 34793 hbexg 40897 |
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