Theorem axc16nf

Description: If dtru 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 . (Revised by Wolf Lammen, 9-Sep-2018) (Proof shortened by BJ, 14-Jun-2019) Remove dependency on ax-10 . (Revised by Wolf Lammen, 12-Oct-2021)

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

Proof

Step	Hyp	Ref	Expression
1		axc16g	|- ( A. x x = y -> ( -. ph -> A. z -. ph ) )
2		eximal	|- ( ( E. z ph -> ph ) <-> ( -. ph -> A. z -. ph ) )
3	1 2	sylibr	|- ( A. x x = y -> ( E. z ph -> ph ) )
4		axc16g	|- ( A. x x = y -> ( ph -> A. z ph ) )
5	3 4	syld	|- ( A. x x = y -> ( E. z ph -> A. z ph ) )
6	5	nfd	|- ( A. x x = y -> F/ z ph )