Theorem nfcrd 2919

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 2915	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 18	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-clel 2838  df-nfc 2912

This theorem is referenced by:  nfeld  2936  dvelimdc  2949  nfraldw  3310  nfcsbd  3878  nfcsbw  3879  nfifd  4517  nfdisjw  5088  axextnd  10571  axrepndlem1  10572  axunndlem1  10575  axregnd  10584  nfchnd  18662  axsepg3  35554  axsepg3ALT  35555  axsepg5  35557  axextdist  36289  nfintd  50451  nfiund  50452  nfiundg  50453