Theorem ax6blem 1580

Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1581 compared to hbnt 1583. (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 1524	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
4	3	con3i 594	. 2 ⊢ (¬ 𝜑 → ¬ ∃𝑥𝜑)
5		alnex 1428	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	4, 5	sylibr 132	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  ax6b  1581