Theorem nfcrd 2890

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

Hypothesis

Ref	Expression
nfcrd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfcrd	⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfcrd

Step	Hyp	Ref	Expression
1		nfcrd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		nfcr 2886	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-clel 2809  df-nfc 2883

This theorem is referenced by:  nfeld  2908  dvelimdc  2921  nfraldw  3279  nfcsbd  3872  nfcsbw  3873  nfifd  4507  axextnd  10500  axrepndlem1  10501  axunndlem1  10504  axregnd  10513  nfchnd  18532  axsepg2  35187  axsepg2ALT  35188  axextdist  35940  nfintd  49860  nfiund  49861  nfiundg  49862