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Mirrors > Home > ILE Home > Th. List > nf5-1 | GIF version |
Description: One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.) |
Ref | Expression |
---|---|
nf5-1 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1579 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)) | |
2 | hbe1a 2003 | . . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | syl6 33 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 3 | nfd2 2002 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 Ⅎwnf 1440 ∃wex 1472 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: nf5d 2005 |
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