Theorem nfcvf 2342

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 2319	. 2 |- F/_ x z
2		nfcv 2319	. 2 |- F/_ z y
3		id 19	. 2 |- ( z = y -> z = y )
4	1, 2, 3	dvelimc 2341	1 |- ( -. A. x x = y -> F/_ x y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1351   F/_wnfc 2306

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  nfcvf2  2343