Theorem nfcvf 2331

Description: If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)

Assertion

Ref	Expression
nfcvf	|- ( -. A. x x = y -> F/_ x y )

Proof of Theorem nfcvf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2308	. 2 |- F/_ x z
2		nfcv 2308	. 2 |- F/_ z y
3		id 19	. 2 |- ( z = y -> z = y )
4	1, 2, 3	dvelimc 2330	1 |- ( -. A. x x = y -> F/_ x y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1341   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfcvf2  2332