Theorem nfcrd 2898

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 2894	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891

This theorem is referenced by:  nfeld  2916  dvelimdc  2929  nfraldw  3308  nfcsbd  3923  nfcsbw  3924  nfifd  4554  axextnd  10632  axrepndlem1  10633  axunndlem1  10636  axregnd  10645  axsepg2  35097  axsepg2ALT  35098  axextdist  35801  nfintd  49247  nfiund  49248  nfiundg  49249