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Mirrors > Home > ILE Home > Th. List > nfcr | GIF version |
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcr | ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2167 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
2 | sp 1401 | . 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylbi 114 | 1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 ∈ wcel 1393 Ⅎwnfc 2165 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-nfc 2167 |
This theorem is referenced by: nfcrii 2171 nfcrd 2191 abidnf 2709 csbtt 2862 csbnestgf 2898 |
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