Theorem nfcvf 2354

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 2331	. 2 |- F/_ x z
2		nfcv 2331	. 2 |- F/_ z y
3		id 19	. 2 |- ( z = y -> z = y )
4	1, 2, 3	dvelimc 2353	1 |- ( -. A. x x = y -> F/_ x y )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1361   F/_wnfc 2318

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-cleq 2181  df-clel 2184  df-nfc 2320

This theorem is referenced by:  nfcvf2  2355