Theorem nfrald

Description: Deduction version of nfral . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfraldw when possible. (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfrald.1	⊢ Ⅎ 𝑦 𝜑
2		nfrald.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3		nfrald.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
5		nfcvf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )
6	5	adantl	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑦 )
7	2	adantr	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝐴 )
8	6 7	nfeld	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
9	3	adantr	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
10	8 9	nfimd	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
11	1 10	nfald2	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
12	4 11	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 𝜓 )