Theorem nfcriv 2939

Description: Consequence of the not-free predicate, similiar to nfcri 2943. Requires 𝑦 and 𝐴 be disjoint, but is not based on ax-13 2344. (Contributed by Wolf Lammen, 13-May-2023.)

Hypothesis

Ref	Expression
nfcriv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcriv	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcriv

Step	Hyp	Ref	Expression
1		nfcriv.1	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcr 2938	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Syntax hints:  Ⅎwnf 1765   ∈ wcel 2081  Ⅎwnfc 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-ex 1762  df-nfc 2935

This theorem is referenced by:  nfcrii  2942  nfnfc  2959  cleqf  2978  nfccdeq  3703  csbgfi  3829  dfss2f  3880  iunxsngf  4913  fedgmullem2  30630