Theorem nfrald 3370

Description: Deduction version of nfral 3372. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfraldw 3307 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 3060	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfrald.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 2930	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfrald.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2915	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfrald.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfimd 1892	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
11	2, 10	nfald2 2448	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
12	1, 11	nfxfrd 1851	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060

This theorem is referenced by:  nfrexd  3371  nfral  3372