Theorem ax6blem 1643

Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1644 compared to hbnt 1646. (Contributed by GD, 27-Jan-2018.)

Hypothesis

Ref	Expression
ax6blem.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
ax6blem	⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem ax6blem

Step	Hyp	Ref	Expression
1		ax6blem.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		id 19	. . . 4 ⊢ (𝜑 → 𝜑)
3	1, 2	exlimih 1586	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	con3i 627	. 2 ⊢ (¬ 𝜑 → ¬ ∃𝑥𝜑)
5		alnex 1492	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	4, 5	sylibr 133	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1346  ∃wex 1485

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  ax6b  1644