Theorem nfaldOLD 2152

Description: Obsolete proof of nfald 2151 as of 16-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfald.1	⊢ Ⅎ𝑦𝜑
nfald.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfaldOLD	⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Proof of Theorem nfaldOLD

Step	Hyp	Ref	Expression
1		nfald.1	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfald.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	1, 2	alrimi 2069	. 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓)
4		nfnf1 2018	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
5	4	nfal 2139	. . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓
6		hba1 2137	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓)
7		sp 2041	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
8	7	nf5rd 2054	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
9	6, 8	hbald 2028	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
10	5, 9	nf5d 2104	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓)
11	3, 10	syl 17	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

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-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)