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 x x = y z φ

Proof

Step Hyp Ref Expression
1 axc16g x x = y ¬ φ z ¬ φ
2 eximal z φ φ ¬ φ z ¬ φ
3 1 2 sylibr x x = y z φ φ
4 axc16g x x = y φ z φ
5 3 4 syld x x = y z φ z φ
6 5 nfd x x = y z φ