Theorem bj-nfexd

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

	Ref	Expression
Hypotheses	bj-nfald.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-nfald.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	bj-nfexd	⊢ ( 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		df-ex	⊢ ( ∃ 𝑦 𝜓 ↔ ¬ ∀ 𝑦 ¬ 𝜓 )
4	2	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝜓 )
5	1 4	bj-nfald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ¬ 𝜓 )
6	5	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ ∀ 𝑦 ¬ 𝜓 )
7	3 6	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )