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Mirrors > Home > ILE Home > Th. List > nf3 | GIF version |
Description: An alternate definition of df-nf 1454. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nf3 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf2 1661 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | nfe1 1489 | . . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | 2 | nfri 1512 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
4 | 3 | 19.21h 1550 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
5 | 1, 4 | bitr4i 186 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 Ⅎwnf 1453 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: hbe1a 2016 eusv2nf 4439 |
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