Theorem nfrexdg

Description: Deduction version of nfrexg . Usage of this theorem is discouraged because it depends on ax-13 . See nfrexd for a version with a disjoint variable condition, but not requiring ax-13 . (Contributed by Mario Carneiro, 14-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfrexdg.1	⊢ Ⅎ 𝑦 𝜑
	nfrexdg.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfrexdg.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfrexdg	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfrexdg.1	⊢ Ⅎ 𝑦 𝜑
2		nfrexdg.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3		nfrexdg.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4		dfrex2	⊢ ( ∃ 𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
5	3	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝜓 )
6	1 2 5	nfrald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
7	6	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
8	4 7	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜓 )