Theorem hbntg 32659

Description: A more general form of hbnt 2268. (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 2299	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜓 → 𝜓)
2	1	con1i 149	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ ∀𝑥𝜓)
3		con3 156	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (¬ ∀𝑥𝜓 → ¬ 𝜑))
4	3	al2imi 1797	. 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 1520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-ex 1762

This theorem is referenced by:  hbimtg  32660  hbng  32662