Theorem hbn 1703

Description: If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. This theorem does not depend on ax-ial 1587, contrary to hbn1 1704 and hbnt 1705. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-ial 1587. (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 1646	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	con3i 641	. 2 ⊢ (¬ 𝜑 → ¬ ∃𝑥𝜑)
5		alnex 1552	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	4, 5	sylibr 134	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1400  ∃wex 1545

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is referenced by:  hbn1  1704  hbnae  1773  sbn  2012  euor  2112  euor2  2145