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Mirrors > Home > ILE Home > Th. List > nfd2 | GIF version |
Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.) |
Ref | Expression |
---|---|
nfd2.1 | ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
nfd2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfd2.1 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | |
2 | nf2 1648 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | sylibr 133 | 1 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
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-gen 1429 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: nf5-1 2004 |
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