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Mirrors > Home > MPE Home > Th. List > nf6 | Structured version Visualization version GIF version |
Description: An alternate definition of df-nf 1791. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nf6 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1791 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfe1 2155 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | 2 | 19.21 2209 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | bitr4i 281 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-ex 1787 df-nf 1791 |
This theorem is referenced by: eusv2nf 5262 xfree 30379 |
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