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Mirrors > Home > MPE Home > Th. List > nfcr | Structured version Visualization version GIF version |
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2171 but use ax-8 2108, df-clel 2816, and avoid a DV condition on 𝑦, 𝐴. (Revised by SN, 3-Jun-2024.) |
Ref | Expression |
---|---|
nfcr | ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2889 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | |
2 | eleq1w 2821 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
3 | 2 | nfbidv 1925 | . . 3 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴)) |
4 | 3 | spvv 2000 | . 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | 1, 4 | sylbi 216 | 1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-nf 1787 df-clel 2816 df-nfc 2889 |
This theorem is referenced by: nfcri 2894 nfcrd 2896 abidnf 3638 csbtt 3849 csbnestgfw 4353 csbnestgf 4358 |
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