Theorem nfcrd 2503

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

Hypothesis

Ref	Expression
nfeqd.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		nfeqd.1	. 2 ⊢ (φ → ℲxA)
2		nfcr 2482	. 2 ⊢ (ℲxA → Ⅎx y ∈ A)
3	1, 2	syl 15	1 ⊢ (φ → Ⅎx y ∈ A)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-ex 1542  df-nfc 2479

This theorem is used by:  nfeqd  2504  nfeld  2505  dvelimdc  2510  nfcsbd  3170  nfifd  3686