Theorem nfimdOLD 1809

Description: Obsolete proof of nfimd 1808 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfimdOLD

Step	Hyp	Ref	Expression
1		nfimd.1	. 2 ⊢ (φ → Ⅎxψ)
2		nfimd.2	. 2 ⊢ (φ → Ⅎxχ)
3		nfa1 1788	. . . . 5 ⊢ Ⅎx∀x(ψ → ∀xψ)
4		hbnt 1775	. . . . . 6 ⊢ (∀x(ψ → ∀xψ) → (¬ ψ → ∀x ¬ ψ))
5		pm2.21 100	. . . . . . . . . . 11 ⊢ (¬ ψ → (ψ → χ))
6	5	alimi 1559	. . . . . . . . . 10 ⊢ (∀x ¬ ψ → ∀x(ψ → χ))
7	6	imim2i 13	. . . . . . . . 9 ⊢ ((¬ ψ → ∀x ¬ ψ) → (¬ ψ → ∀x(ψ → χ)))
8	7	adantr 451	. . . . . . . 8 ⊢ (((¬ ψ → ∀x ¬ ψ) ∧ (χ → ∀xχ)) → (¬ ψ → ∀x(ψ → χ)))
9		ax-1 6	. . . . . . . . . . 11 ⊢ (χ → (ψ → χ))
10	9	alimi 1559	. . . . . . . . . 10 ⊢ (∀xχ → ∀x(ψ → χ))
11	10	imim2i 13	. . . . . . . . 9 ⊢ ((χ → ∀xχ) → (χ → ∀x(ψ → χ)))
12	11	adantl 452	. . . . . . . 8 ⊢ (((¬ ψ → ∀x ¬ ψ) ∧ (χ → ∀xχ)) → (χ → ∀x(ψ → χ)))
13	8, 12	jad 154	. . . . . . 7 ⊢ (((¬ ψ → ∀x ¬ ψ) ∧ (χ → ∀xχ)) → ((ψ → χ) → ∀x(ψ → χ)))
14	13	ex 423	. . . . . 6 ⊢ ((¬ ψ → ∀x ¬ ψ) → ((χ → ∀xχ) → ((ψ → χ) → ∀x(ψ → χ))))
15	4, 14	syl 15	. . . . 5 ⊢ (∀x(ψ → ∀xψ) → ((χ → ∀xχ) → ((ψ → χ) → ∀x(ψ → χ))))
16	3, 15	alimd 1764	. . . 4 ⊢ (∀x(ψ → ∀xψ) → (∀x(χ → ∀xχ) → ∀x((ψ → χ) → ∀x(ψ → χ))))
17	16	imp 418	. . 3 ⊢ ((∀x(ψ → ∀xψ) ∧ ∀x(χ → ∀xχ)) → ∀x((ψ → χ) → ∀x(ψ → χ)))
18		df-nf 1545	. . . 4 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
19		df-nf 1545	. . . 4 ⊢ (Ⅎxχ ↔ ∀x(χ → ∀xχ))
20	18, 19	anbi12i 678	. . 3 ⊢ ((Ⅎxψ ∧ Ⅎxχ) ↔ (∀x(ψ → ∀xψ) ∧ ∀x(χ → ∀xχ)))
21		df-nf 1545	. . 3 ⊢ (Ⅎx(ψ → χ) ↔ ∀x((ψ → χ) → ∀x(ψ → χ)))
22	17, 20, 21	3imtr4i 257	. 2 ⊢ ((Ⅎxψ ∧ Ⅎxχ) → Ⅎx(ψ → χ))
23	1, 2, 22	syl2anc 642	1 ⊢ (φ → Ⅎx(ψ → χ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)