Theorem nfcd 2485

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 1765	. 2 ⊢ (φ → ∀yℲx y ∈ A)
4		df-nfc 2479	. 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A)
5	3, 4	sylibr 203	1 ⊢ (φ → ℲxA)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2477

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-nf 1545  df-nfc 2479

This theorem is referenced by:  nfabd2  2508  dvelimdc  2510  sbnfc2  3197