Theorem a16nf 1854

Description: If there is only one element in the universe, then everything satisfies F/. (Contributed by Mario Carneiro, 7-Oct-2016.)

Assertion

Ref	Expression
a16nf	|- ( A. x x = y -> F/ z ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem a16nf

Step	Hyp	Ref	Expression
1		nfae 1707	. 2 |- F/ z A. x x = y
2		a16g 1852	. 2 |- ( A. x x = y -> ( ph -> A. z ph ) )
3	1, 2	nfd 1511	1 |- ( A. x x = y -> F/ z ph )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448

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-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-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  nfsbxy  1930  nfsbxyt  1931  dvelimor  2006