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 → Ⅎ 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 → Ⅎ 𝑧 𝜑 )