Theorem a16nf 1818

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 1678	. 2 |- F/ z A. x x = y
2		a16g 1816	. 2 |- ( A. x x = y -> ( ph -> A. z ph ) )
3	1, 2	nfd 1484	1 |- ( A. x x = y -> F/ z ph )

Syntax hints:   -> wi 4   A.wal 1310   F/wnf 1417

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717

This theorem is referenced by:  nfsbxy  1891  nfsbxyt  1892  dvelimor  1967