Theorem nfcr 2481

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

Assertion

Ref	Expression
nfcr	⊢ (ℲxA → Ⅎx y ∈ A)

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem nfcr

Step	Hyp	Ref	Expression
1		df-nfc 2478	. 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A)
2		sp 1747	. 2 ⊢ (∀yℲx y ∈ A → Ⅎx y ∈ A)
3	1, 2	sylbi 187	1 ⊢ (ℲxA → Ⅎx y ∈ A)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2476

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542  df-nfc 2478

This theorem is referenced by:  nfcrii  2482  nfcrd  2502  abidnf  3005  csbtt  3148  csbnestgf  3184