Theorem a16nf 1890

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 1743	. 2 |- F/ z A. x x = y
2		a16g 1888	. 2 |- ( A. x x = y -> ( ph -> A. z ph ) )
3	1, 2	nfd 1547	1 |- ( A. x x = y -> F/ z ph )

Syntax hints:   -> wi 4   A.wal 1371   F/wnf 1484

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  nfsbxy  1971  nfsbxyt  1972  dvelimor  2047