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Mirrors > Home > ILE Home > Th. List > nfcd | 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 1522 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
4 | df-nfc 2308 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
5 | 3, 4 | sylibr 134 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 Ⅎwnf 1460 ∈ wcel 2148 Ⅎwnfc 2306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-nfc 2308 |
This theorem is referenced by: nfabdw 2338 nfabd 2339 dvelimdc 2340 sbnfc2 3117 |
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