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Mirrors > Home > ILE Home > Th. List > nfd | GIF version |
Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nfd.1 | ⊢ Ⅎ𝑥𝜑 |
nfd.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
nfd | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1452 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfd.2 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
4 | 2, 3 | alrimih 1398 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)) |
5 | df-nf 1390 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | sylibr 132 | 1 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 Ⅎwnf 1389 |
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-4 1440 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: nfdh 1457 nfrimi 1458 nfnt 1586 cbv1h 1673 nfald 1683 a16nf 1787 dvelimALT 1927 dvelimfv 1928 nfsb4t 1931 hbeud 1963 |
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