Theorem hbia1 1605

Description: Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)

Assertion

Ref	Expression
hbia1	⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem hbia1

Step	Hyp	Ref	Expression
1		hba1 1593	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2		hba1 1593	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
3	1, 2	hbim 1598	1 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1500  ax-gen 1502  ax-4 1563  ax-ial 1587  ax-i5r 1588

This theorem is used by: (None)