Theorem axc16nf 2027

Description: If dtru 4594 is false, then there is only one element in the universe, so everything satisfies F/. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 1920. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.)

Assertion

Ref	Expression
axc16nf	|- ( A. x x = y -> F/ z ph )

Distinct variable group:   x, y

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

Proof of Theorem axc16nf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 2026	. 2 |- ( A. x x = y -> A. z z = w )
2		nfa1 1979	. . 3 |- F/ z A. z z = w
3		axc16 2024	. . 3 |- ( A. z z = w -> ( ph -> A. z ph ) )
4	2, 3	nfd 1956	. 2 |- ( A. z z = w -> F/ z ph )
5	1, 4	syl 17	1 |- ( A. x x = y -> F/ z ph )

Syntax hints:   -> wi 4   A.wal 1442   F/wnf 1667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668

This theorem is referenced by:  nfsb  2269  nfsbd  2271