Theorem nfrald 3150

Description: Deduction version of nfral 3153. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfraldw 3148 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrald.1	⊢ Ⅎ𝑦𝜑
nfrald.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfrald.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrald	⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Proof of Theorem nfrald

Step	Hyp	Ref	Expression
1		df-ral 3069	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfrald.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 2936	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfrald.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2918	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfrald.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfimd 1897	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
11	2, 10	nfald2 2445	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
12	1, 11	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069

This theorem is referenced by:  nfral  3153  nfrexdg  3241