Theorem nfrald 2666

Description: Deduction version of nfral 2668. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfrald.2	⊢ Ⅎyφ
nfrald.3	⊢ (φ → ℲxA)
nfrald.4	⊢ (φ → Ⅎxψ)

Assertion

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

Proof of Theorem nfrald

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀y ∈ A ψ ↔ ∀y(y ∈ A → ψ))
2		nfrald.2	. . 3 ⊢ Ⅎyφ
3		nfcvf 2512	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎxy)
4	3	adantl 452	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxy)
5		nfrald.3	. . . . . 6 ⊢ (φ → ℲxA)
6	5	adantr 451	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → ℲxA)
7	4, 6	nfeld 2505	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx y ∈ A)
8		nfrald.4	. . . . 5 ⊢ (φ → Ⅎxψ)
9	8	adantr 451	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
10	7, 9	nfimd 1808	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx(y ∈ A → ψ))
11	2, 10	nfald2 1972	. 2 ⊢ (φ → Ⅎx∀y(y ∈ A → ψ))
12	1, 11	nfxfrd 1571	1 ⊢ (φ → Ⅎx∀y ∈ A ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is used by:  nfrexd  2667  nfral  2668