Theorem hba1 2289

Description: The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
hba1	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Proof of Theorem hba1

Step	Hyp	Ref	Expression
1		nfa1 2148	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2188	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786

This theorem is referenced by:  axi5r  2695  axial  2696  bj-19.41al  35524  bj-wnf1  35583  hbntal  43299  hbimpg  43300  hbimpgVD  43650  hbalgVD  43651  hbexgVD  43652  ax6e2eqVD  43653  e2ebindVD  43658  vk15.4jVD  43660