Theorem nfcr 2888

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 2810, and avoid a DV condition on 𝑦, 𝐴. (Revised by SN, 3-Jun-2024.)

Assertion

Ref	Expression
nfcr	⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2885	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		eleq1w 2816	. . . 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)

Syntax hints:   → wi 4  ∀wal 1539  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 1782  df-nf 1786  df-clel 2810  df-nfc 2885

This theorem is referenced by:  nfcri  2890  nfcrd  2892  abidnf  3698  csbtt  3910  csbnestgfw  4419  csbnestgf  4424