Theorem nfcr 2756

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 2753	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		sp 2053	. 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	sylbi 207	1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nfc 2753

This theorem is referenced by:  nfcrii  2757  nfcrd  2771  nfnfc  2774  abidnf  3375  csbtt  3544  csbnestgf  3996  bj-nfcrii  32851