Theorem hbimtg 33782

Description: A more general and closed form of hbim 2296. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
hbimtg	⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))

Proof of Theorem hbimtg

Step	Hyp	Ref	Expression
1		hbntg 33781	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥 ¬ 𝜑))
2		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜃))
3	2	alimi 1814	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜃))
4	1, 3	syl6 35	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥(𝜑 → 𝜃)))
5	4	adantr 481	. 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (¬ 𝜒 → ∀𝑥(𝜑 → 𝜃)))
6		ala1 1816	. . . 4 ⊢ (∀𝑥𝜃 → ∀𝑥(𝜑 → 𝜃))
7	6	imim2i 16	. . 3 ⊢ ((𝜓 → ∀𝑥𝜃) → (𝜓 → ∀𝑥(𝜑 → 𝜃)))
8	7	adantl 482	. 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (𝜓 → ∀𝑥(𝜑 → 𝜃)))
9	5, 8	jad 187	1 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  hbimg  33785