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Theorem nfcr 2884
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2178 but use ax-8 2115, df-clel 2811, 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 2881 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 eleq1w 2815 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
32nfbidv 1928 . . 3 (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴))
43spvv 2007 . 2 (∀𝑧𝑥 𝑧𝐴 → Ⅎ𝑥 𝑦𝐴)
51, 4sylbi 220 1 (𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wnf 1790  wcel 2113  wnfc 2879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-nf 1791  df-clel 2811  df-nfc 2881
This theorem is referenced by:  nfcri  2886  nfcrd  2888  abidnf  3603  csbtt  3808  csb0  4297  csbnestgfw  4310  csbnestgf  4315
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