Theorem nfcrd 2971

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 groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem nfcrd

Step	Hyp	Ref	Expression
1		nfcrd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		nfcr 2968	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2963

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 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nfc 2965

This theorem is referenced by:  nfeqd  2990  nfeld  2991  dvelimdc  3007  nfcsbd  3910  nfcsbw  3911  nfifd  4497  axextnd  10015  axrepndlem1  10016  axunndlem1  10019  axregnd  10028  axextdist  33046  wl-clelsb3df  34865  nfintd  44783  nfiund  44784  nfiundg  44785