Theorem nfcriv 2966

Description: Consequence of the not-free predicate, similiar to nfcri 2970. Requires 𝑦 and 𝐴 be disjoint, but is not based on ax-13 2389. (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 2965	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Syntax hints:  Ⅎwnf 1783   ∈ wcel 2113  Ⅎwnfc 2960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780  df-nfc 2962

This theorem is referenced by:  nfcrii  2969  nfnfc  2989  cleqf  3009  nfccdeq  3765  csbgfi  3896  dfss2f  3951  iunxsngf  5007  fedgmullem2  31050