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Mirrors > Home > MPE Home > Th. List > nf5 | Structured version Visualization version GIF version |
Description: Alternate definition of df-nf 1701. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
nf5 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1701 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfa1 2015 | . . 3 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
3 | 2 | 19.23 2067 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
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-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nfnf1OLD 2145 drnf1 2317 axie2 2585 xfree 28687 bj-nfdt0 31872 bj-nfalt 31889 bj-nfext 31890 bj-nfs1t 31901 bj-drnf1v 31938 bj-sbnf 32016 wl-sbnf1 32515 hbexg 37793 |
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