Theorem hba1 2283

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 2141	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2184	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  axi5r  2691  axial  2692  bj-19.41al  36130  bj-wnf1  36189  hbntal  43983  hbimpg  43984  hbimpgVD  44334  hbalgVD  44335  hbexgVD  44336  ax6e2eqVD  44337  e2ebindVD  44342  vk15.4jVD  44344