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| Mirrors > Home > MPE Home > Th. List > nf5 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of df-nf 1784. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1784 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
| Ref | Expression |
|---|---|
| nf5 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-nf 1784 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
| 2 | nfa1 2151 | . . 3 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
| 3 | 2 | 19.23 2211 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: sbnfOLD 2313 drnf1vOLD 2376 drnf1 2448 axie2 2703 xfree 32463 bj-nfdt0 36696 bj-nfalt 36712 bj-nfext 36713 bj-nfs1t 36791 wl-sbnf1 37556 hbexg 44576 |
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