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Theorem nfcr 2892
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.)
Assertion
Ref Expression
nfcr (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2889 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 eleq1w 2821 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
32nfbidv 1925 . . 3 (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴))
43spvv 2000 . 2 (∀𝑧𝑥 𝑧𝐴 → Ⅎ𝑥 𝑦𝐴)
51, 4sylbi 216 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wnf 1786  wcel 2106  wnfc 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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