Theorem cbv3hvOLD 1851

Description: Obsolete proof of cbv3hv 1850 as of 29-Dec-2017. (Contributed by NM, 25-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv3hv.1	⊢ (φ → ∀yφ)
cbv3hv.2	⊢ (ψ → ∀xψ)
cbv3hv.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbv3hvOLD	⊢ (∀xφ → ∀yψ)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbv3hvOLD

Step	Hyp	Ref	Expression
1		cbv3hv.1	. . 3 ⊢ (φ → ∀yφ)
2	1	alimi 1559	. 2 ⊢ (∀xφ → ∀x∀yφ)
3		ax9v 1655	. . . . 5 ⊢ ¬ ∀x ¬ x = y
4		hba1 1786	. . . . . . . 8 ⊢ (∀xφ → ∀x∀xφ)
5		cbv3hv.2	. . . . . . . 8 ⊢ (ψ → ∀xψ)
6	4, 5	hbim 1817	. . . . . . 7 ⊢ ((∀xφ → ψ) → ∀x(∀xφ → ψ))
7	6	hbn 1776	. . . . . 6 ⊢ (¬ (∀xφ → ψ) → ∀x ¬ (∀xφ → ψ))
8		sp 1747	. . . . . . . 8 ⊢ (∀xφ → φ)
9		cbv3hv.3	. . . . . . . 8 ⊢ (x = y → (φ → ψ))
10	8, 9	syl5 28	. . . . . . 7 ⊢ (x = y → (∀xφ → ψ))
11	10	con3i 127	. . . . . 6 ⊢ (¬ (∀xφ → ψ) → ¬ x = y)
12	7, 11	alrimih 1565	. . . . 5 ⊢ (¬ (∀xφ → ψ) → ∀x ¬ x = y)
13	3, 12	mt3 171	. . . 4 ⊢ (∀xφ → ψ)
14	13	alimi 1559	. . 3 ⊢ (∀y∀xφ → ∀yψ)
15	14	a7s 1735	. 2 ⊢ (∀x∀yφ → ∀yψ)
16	2, 15	syl 15	1 ⊢ (∀xφ → ∀yψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

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)