Theorem nfaldOLD 1853

Description: Obsolete proof of nfald 1852 as of 6-Jan-2018. (Contributed by Mario Carneiro, 24-Sep-2016.) (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 1765	. 2 ⊢ (φ → ∀yℲxψ)
4		nfnf1 1790	. . . 4 ⊢ ℲxℲxψ
5	4	nfal 1842	. . 3 ⊢ Ⅎx∀yℲxψ
6		nfr 1761	. . . . 5 ⊢ (Ⅎxψ → (ψ → ∀xψ))
7	6	al2imi 1561	. . . 4 ⊢ (∀yℲxψ → (∀yψ → ∀y∀xψ))
8		ax-7 1734	. . . 4 ⊢ (∀y∀xψ → ∀x∀yψ)
9	7, 8	syl6 29	. . 3 ⊢ (∀yℲxψ → (∀yψ → ∀x∀yψ))
10	5, 9	nfd 1766	. 2 ⊢ (∀yℲxψ → Ⅎx∀yψ)
11	3, 10	syl 15	1 ⊢ (φ → Ⅎx∀yψ)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from 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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)