Theorem hbnt 1641

Description: Closed theorem version of bound-variable hypothesis builder hbn 1642. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Assertion

Ref	Expression
hbnt	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		ax-4 1498	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
2	1	con3i 622	. . 3 ⊢ (¬ 𝜑 → ¬ ∀𝑥𝜑)
3		ax6b 1639	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4	2, 3	syl 14	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5		con3 632	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
6	5	al2imi 1446	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
7	4, 6	syl5 32	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349

This theorem is referenced by:  hbn  1642  hbnd  1643  nfnt  1644