Theorem nfcd 2345

Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfcd.1	⊢ Ⅎ𝑦𝜑
nfcd.2	⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
nfcd	⊢ (𝜑 → Ⅎ𝑥𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem nfcd

Step	Hyp	Ref	Expression
1		nfcd.1	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfcd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	alrimi 1546	. 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
4		df-nfc 2339	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
5	3, 4	sylibr 134	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484   ∈ wcel 2178  Ⅎwnfc 2337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-4 1534

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-nfc 2339

This theorem is referenced by:  nfabdw  2369  nfabd  2370  dvelimdc  2371  sbnfc2  3162