Theorem nfcr 2894

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2176 but use ax-8 2109, df-clel 2815, 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 2891	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		eleq1w 2823	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	nfbidv 1921	. . 3 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴))
4	3	spvv 1995	. 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5	1, 4	sylbi 217	1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891

This theorem is referenced by:  nfcri  2896  nfcrd  2898  abidnf  3707  csbtt  3915  csbnestgfw  4421  csbnestgf  4426