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	$$⊢\forall xx=y\to Ⅎz\phi$$

Proof

Step	Hyp	Ref	Expression
1		axc16g	$$⊢\forall xx=y\to \left(\neg \phi \to \forall z\neg \phi \right)$$
2		eximal	$$⊢\left(\exists z\phi \to \phi \right)\leftrightarrow \left(\neg \phi \to \forall z\neg \phi \right)$$
3	1 2	sylibr	$$⊢\forall xx=y\to \left(\exists z\phi \to \phi \right)$$
4		axc16g	$$⊢\forall xx=y\to \left(\phi \to \forall z\phi \right)$$
5	3 4	syld	$$⊢\forall xx=y\to \left(\exists z\phi \to \forall z\phi \right)$$
6	5	nfd	$$⊢\forall xx=y\to Ⅎz\phi$$