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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	$⊢ \forall x x = y \to Ⅎ z φ$

Proof

Step	Hyp	Ref	Expression
1		axc16g	$⊢ \forall x x = y \to (\neg φ \to \forall z \neg φ)$
2		eximal	$⊢ (\exists z φ \to φ) \leftrightarrow (\neg φ \to \forall z \neg φ)$
3	1 2	sylibr	$⊢ \forall x x = y \to (\exists z φ \to φ)$
4		axc16g	$⊢ \forall x x = y \to (φ \to \forall z φ)$
5	3 4	syld	$⊢ \forall x x = y \to (\exists z φ \to \forall z φ)$
6	5	nfd	$⊢ \forall x x = y \to Ⅎ z φ$