Theorem nfcrd 2361

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfeqd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfcrd	⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfcrd

Step	Hyp	Ref	Expression
1		nfeqd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		nfcr 2339	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 14	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1482   ∈ wcel 2175  Ⅎwnfc 2334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1532

This theorem depends on definitions:  df-bi 117  df-nfc 2336

This theorem is referenced by:  nfeqd  2362  nfeld  2363  dvelimdc  2368  nfcsbd  3128  nfcsbw  3129  nfifd  3597