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| Mirrors > Home > MPE Home > Th. List > nf6 | Structured version Visualization version GIF version | ||
| Description: An alternate definition of df-nf 1784. (Contributed by Mario Carneiro, 24-Sep-2016.) | 
| Ref | Expression | 
|---|---|
| nf6 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-nf 1784 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
| 2 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
| 3 | 2 | 19.21 2207 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | 
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | 
| 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-ex 1780 df-nf 1784 | 
| This theorem is referenced by: eusv2nf 5395 xfree 32463 | 
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