Theorem nfci 2480

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

Hypothesis

Ref	Expression
nfci.1	⊢ Ⅎx y ∈ A

Assertion

Ref	Expression
nfci	⊢ ℲxA

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem nfci

Step	Hyp	Ref	Expression
1		df-nfc 2479	. 2 ⊢ (ℲxA ↔ ∀yℲx y ∈ A)
2		nfci.1	. 2 ⊢ Ⅎx y ∈ A
3	1, 2	mpgbir 1550	1 ⊢ ℲxA

Syntax hints:  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

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

This theorem is referenced by:  nfcii  2481  nfcv  2490  nfab1  2492  nfab  2494