Theorem nfcr 2917

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2216 but use ax-8 2148, df-clel 2840, 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 2914	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		eleq1w 2848	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	nfbidv 1955	. . 3 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴))
4	3	spvv 2021	. 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5	1, 4	sylbi 220	1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2840  df-nfc 2914

This theorem is used by:  nfcri  2919  nfcrd  2921  abidnf  3667  csbtt  3871  csbnestgfw  4387  csbnestgf  4392  nfchnd  18691