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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 2213 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
4 | df-nfc 2965 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 |
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-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 df-nfc 2965 |
This theorem is referenced by: nfabdw 3002 nfabd 3003 nfabd2OLD 3005 dvelimdc 3007 nfcvf 3009 sbnfc2 4390 |
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