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Mirrors > Home > MPE Home > Th. List > nf5d | Structured version Visualization version GIF version |
Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nf5d.1 | ⊢ Ⅎ𝑥𝜑 |
nf5d.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
nf5d | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5d.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nf5d.2 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | alrimi 2213 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)) |
4 | nf5-1 2149 | . 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 |
This theorem is referenced by: dvelimhw 2366 nfeqf 2399 cbv1h 2425 axc16nfALT 2459 nfsb2 2522 nfsb2ALT 2600 nfabdw 3002 distel 33050 bj-cbv1hv 34120 wl-ax11-lem3 34821 |
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