Theorem bj-nfald

Description: Variant of nfald . (Contributed by BJ, 25-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nfald.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
2		bj-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	alimdh	⊢ ( 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 )