Theorem hbnt 1543

Description: Closed theorem version of bound-variable hypothesis builder hbn 1544. (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 1400	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
2	1	con3i 562	. . 3 ⊢ (¬ 𝜑 → ¬ ∀𝑥𝜑)
3		ax6b 1541	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4	2, 3	syl 14	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
5		con3 571	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
6	5	al2imi 1347	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
7	4, 6	syl5 28	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  hbn  1544  hbnd  1545  nfnt  1546