Theorem hbntg 36659

Description: A more general form of hbnt 2327. (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 2347	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜓 → 𝜓)
2	1	con1i 148	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ ∀𝑥𝜓)
3		con3 154	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (¬ ∀𝑥𝜓 → ¬ 𝜑))
4	3	al2imi 1847	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (∀𝑥 ¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜑))
5	2, 4	syl5 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  hbimtg  36660  hbng  36662