Theorem nfexd

Description: If x is not free in ps , it is not free in E. y ps . (Contributed by Mario Carneiro, 24-Sep-2016)

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

Proof

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