Theorem nfimdOLD 2214

Description: Obsolete proof of nfimd 1812 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfimdOLD.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfimdOLD.2	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
nfimdOLD	⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))

Proof of Theorem nfimdOLD

Step	Hyp	Ref	Expression
1		nfimdOLD.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2		nfimdOLD.2	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
3		nfnf1OLDOLD 2196	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
4		nfnf1OLDOLD 2196	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜒
5		nfrOLD 2176	. . . . . 6 ⊢ (Ⅎ𝑥𝜒 → (𝜒 → ∀𝑥𝜒))
6	5	imim2d 55	. . . . 5 ⊢ (Ⅎ𝑥𝜒 → ((𝜓 → 𝜒) → (𝜓 → ∀𝑥𝜒)))
7		19.21tOLD 2201	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → 𝜒) ↔ (𝜓 → ∀𝑥𝜒)))
8	7	biimprd 237	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → ((𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓 → 𝜒)))
9	6, 8	syl9r 76	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))))
10	3, 4, 9	alrimdOLD 2184	. . 3 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ∀𝑥((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))))
11		df-nfOLD 1712	. . 3 ⊢ (Ⅎ𝑥(𝜓 → 𝜒) ↔ ∀𝑥((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
12	10, 11	syl6ibr 241	. 2 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝜓 → 𝜒)))
13	1, 2, 12	sylc 63	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1473  ℲwnfOLD 1700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-nfOLD 1712

This theorem is referenced by:  hbimdOLD  2218  nfandOLD  2220  nfbidOLD  2230