Theorem nfcvf 2395

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 2372	. 2 |- F/_ x z
2		nfcv 2372	. 2 |- F/_ z y
3		id 19	. 2 |- ( z = y -> z = y )
4	1, 2, 3	dvelimc 2394	1 |- ( -. A. x x = y -> F/_ x y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1393   F/_wnfc 2359

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by:  nfcvf2  2396