Theorem nfcrd 2757

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 2743	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nfc 2740

This theorem is referenced by:  nfeqd  2758  nfeld  2759  dvelimdc  2772  nfcsbd  3516  nfifd  4064  axextnd  9292  axrepndlem1  9293  axunndlem1  9296  axregnd  9305  axextdist  30949  nfiund  42219