Metamath Proof Explorer


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 xx=yzφ

Proof

Step Hyp Ref Expression
1 axc16g xx=y¬φz¬φ
2 eximal zφφ¬φz¬φ
3 1 2 sylibr xx=yzφφ
4 axc16g xx=yφzφ
5 3 4 syld xx=yzφzφ
6 5 nfd xx=yzφ