Theorem hba1 2334

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 2192	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2237	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1807  df-nf 1811

This theorem is referenced by:  axi5r  2733  axial  2734  bj-19.41al  37170  bj-wnf1  37233  hbntal  45154  hbimpg  45155  hbimpgVD  45504  hbalgVD  45505  hbexgVD  45506  ax6e2eqVD  45507  e2ebindVD  45512  vk15.4jVD  45514  quantgodelALT  47481