Theorem nfndOLD 1792

Description: Obsolete proof of nfnd 1791 as of 28-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfnd.1	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfndOLD	⊢ (φ → Ⅎx ¬ ψ)

Proof of Theorem nfndOLD

Step	Hyp	Ref	Expression
1		nfnd.1	. 2 ⊢ (φ → Ⅎxψ)
2		nfnf1 1790	. . 3 ⊢ ℲxℲxψ
3		ax6o 1750	. . . . 5 ⊢ (¬ ∀x ¬ ∀xψ → ψ)
4	3	con1i 121	. . . 4 ⊢ (¬ ψ → ∀x ¬ ∀xψ)
5		df-nf 1545	. . . . 5 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
6		con3 126	. . . . . 6 ⊢ ((ψ → ∀xψ) → (¬ ∀xψ → ¬ ψ))
7	6	al2imi 1561	. . . . 5 ⊢ (∀x(ψ → ∀xψ) → (∀x ¬ ∀xψ → ∀x ¬ ψ))
8	5, 7	sylbi 187	. . . 4 ⊢ (Ⅎxψ → (∀x ¬ ∀xψ → ∀x ¬ ψ))
9	4, 8	syl5 28	. . 3 ⊢ (Ⅎxψ → (¬ ψ → ∀x ¬ ψ))
10	2, 9	nfd 1766	. 2 ⊢ (Ⅎxψ → Ⅎx ¬ ψ)
11	1, 10	syl 15	1 ⊢ (φ → Ⅎx ¬ ψ)

Syntax hints:  ¬ wn 3   → 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-11 1746

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

This theorem is referenced by: (None)