Theorem nfcr 2941

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2175 but use ax-8 2113, 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 2938	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		eleq1w 2872	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	nfbidv 1923	. . 3 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴))
4	3	spvv 2003	. 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5	1, 4	sylbi 220	1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936

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 1911  ax-6 1970  ax-7 2015  ax-8 2113

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-clel 2870  df-nfc 2938

This theorem is referenced by:  nfcri  2943  nfcrd  2945  abidnf  3642  csbtt  3845  csb0  4314  csbnestgfw  4327  csbnestgf  4332  wl-dfralf  35004