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Mirrors > Home > MPE Home > Th. List > nfd | Structured version Visualization version GIF version |
Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.) |
Ref | Expression |
---|---|
nfd.1 | ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
nfd | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfd.1 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | |
2 | df-nf 1791 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | sylibr 237 | 1 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-nf 1791 |
This theorem is referenced by: nftht 1799 nfntht 1800 nfimd 1900 nf5-1 2148 axc16nf 2263 nfald 2329 nfeqf2 2376 bj-nfald 34918 |
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