Theorem nfcrd 2895

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 2891	. 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-clel 2817  df-nfc 2888

This theorem is referenced by:  nfeld  2917  dvelimdc  2933  nfraldw  3146  nfcsbd  3854  nfcsbw  3855  nfifd  4485  axextnd  10278  axrepndlem1  10279  axunndlem1  10282  axregnd  10291  axextdist  33681  nfintd  46265  nfiund  46266  nfiundg  46267