Theorem hbntg 35248

Description: A more general form of hbnt 2289. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
hbntg	⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbntg

Step	Hyp	Ref	Expression
1		axc7 2309	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜓 → 𝜓)
2	1	con1i 147	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ ∀𝑥𝜓)
3		con3 153	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (¬ ∀𝑥𝜓 → ¬ 𝜑))
4	3	al2imi 1816	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (∀𝑥 ¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜑))
5	2, 4	syl5 34	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by:  hbimtg  35249  hbng  35251