![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfcd | Structured version Visualization version GIF version |
Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcd.1 | ⊢ Ⅎ𝑦𝜑 |
nfcd.2 | ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
nfcd | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcd.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfcd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | |
3 | 1, 2 | alrimi 2202 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
4 | df-nfc 2881 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 df-nfc 2881 |
This theorem is referenced by: nfabdw 2923 nfabdwOLD 2924 nfabd 2925 dvelimdc 2927 nfcvf 2929 sbnfc2 4437 |
Copyright terms: Public domain | W3C validator |