Theorem nfimdOLDOLD 1824

Description: Obsolete proof of nfimd 1823 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfimdOLDOLD

Step	Hyp	Ref	Expression
1		19.35 1805	. . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → ∃𝑥𝜒))
2		nfimd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nf4 1713	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 ↔ (¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜓))
4	2, 3	sylib 208	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜓))
5		pm2.21 120	. . . . . 6 ⊢ (¬ 𝜓 → (𝜓 → 𝜒))
6	5	alimi 1739	. . . . 5 ⊢ (∀𝑥 ¬ 𝜓 → ∀𝑥(𝜓 → 𝜒))
7	4, 6	syl6 35	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥𝜓 → ∀𝑥(𝜓 → 𝜒)))
8		nfimd.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜒)
9	8	nfrd 1717	. . . . 5 ⊢ (𝜑 → (∃𝑥𝜒 → ∀𝑥𝜒))
10		ala1 1741	. . . . 5 ⊢ (∀𝑥𝜒 → ∀𝑥(𝜓 → 𝜒))
11	9, 10	syl6 35	. . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → ∀𝑥(𝜓 → 𝜒)))
12	7, 11	jad 174	. . 3 ⊢ (𝜑 → ((∀𝑥𝜓 → ∃𝑥𝜒) → ∀𝑥(𝜓 → 𝜒)))
13	1, 12	syl5bi 232	. 2 ⊢ (𝜑 → (∃𝑥(𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
14	13	nfd 1716	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)