Theorem hbn 1699

Description: If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. This theorem does not depend on ax-ial 1583, contrary to hbn1 1700 and hbnt 1701. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-ial 1583. (Revised by GD, 27-Jan-2018.)

Hypothesis

Ref	Expression
hbn.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbn	⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem hbn

Step	Hyp	Ref	Expression
1		hbn.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		id 19	. . . 4 ⊢ (𝜑 → 𝜑)
3	1, 2	exlimih 1642	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	con3i 637	. 2 ⊢ (¬ 𝜑 → ¬ ∃𝑥𝜑)
5		alnex 1548	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	4, 5	sylibr 134	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1396  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie2 1543

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404

This theorem is referenced by:  hbn1  1700  hbnae  1769  sbn  2005  euor  2105  euor2  2138