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Mirrors > Home > ILE Home > Th. List > nf3 | GIF version |
Description: An alternate definition of df-nf 1390. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nf3 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf2 1598 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfe1 1425 | . . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | 2 | nfri 1452 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
4 | 3 | 19.21h 1489 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
5 | 1, 4 | bitr4i 185 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: eusv2nf 4206 |
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