Theorem nfcii 2481

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

Hypothesis

Ref	Expression
nfcii.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
nfcii	⊢ ℲxA

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem nfcii

Step	Hyp	Ref	Expression
1		nfcii.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	nfi 1551	. 2 ⊢ Ⅎx y ∈ A
3	2	nfci 2480	1 ⊢ ℲxA

Syntax hints:   → wi 4  ∀wal 1540   ∈ 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

This theorem depends on definitions:  df-bi 177  df-nf 1545  df-nfc 2479

This theorem is referenced by: (None)