Theorem axi5r 2700

Description: Converse of axc4 2322 (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 2294	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2		hba1 2294	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
3	1, 2	hbim 2300	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
4		sp 2184	. . 3 ⊢ (∀𝑥𝜓 → 𝜓)
5	4	imim2i 16	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → 𝜓))
6	3, 5	alrimih 1824	1 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)