Theorem axi5r 2326

Description: Converse of ax-5o (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axi5r	⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ))

Proof of Theorem axi5r

Step	Hyp	Ref	Expression
1		hba1 1786	. . 3 ⊢ (∀xφ → ∀x∀xφ)
2		hba1 1786	. . 3 ⊢ (∀xψ → ∀x∀xψ)
3	1, 2	hbim 1817	. 2 ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ∀xψ))
4		sp 1747	. . . 4 ⊢ (∀xψ → ψ)
5	4	imim2i 13	. . 3 ⊢ ((∀xφ → ∀xψ) → (∀xφ → ψ))
6	5	alimi 1559	. 2 ⊢ (∀x(∀xφ → ∀xψ) → ∀x(∀xφ → ψ))
7	3, 6	syl 15	1 ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ))

Syntax hints:   → 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-11 1746

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

This theorem is referenced by: (None)