Theorem nfald

Description: Deduction form of nfal . (Contributed by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 16-Oct-2021)

	Ref	Expression
Hypotheses	nfald.1	⊢ Ⅎ 𝑦 𝜑
	nfald.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfald.1	⊢ Ⅎ 𝑦 𝜑
2		nfald.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		19.12	⊢ ( ∃ 𝑥 ∀ 𝑦 𝜓 → ∀ 𝑦 ∃ 𝑥 𝜓 )
4	2	nfrd	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) )
5	1 4	alimd	⊢ ( 𝜑 → ( ∀ 𝑦 ∃ 𝑥 𝜓 → ∀ 𝑦 ∀ 𝑥 𝜓 ) )
6		ax-11	⊢ ( ∀ 𝑦 ∀ 𝑥 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
7	3 5 6	syl56	⊢ ( 𝜑 → ( ∃ 𝑥 ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 ) )
8	7	nfd	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 𝜓 )