Theorem hba1 2189

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, 15-Dec-2017.) (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 2068	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2103	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750

This theorem is referenced by:  nfa1OLD  2195  nfaldOLD  2202  nfa1OLDOLD  2243  axi5r  2623  axial  2624  bj-19.41al  32762  bj-modal4e  32830  hbntal  39086  hbimpg  39087  hbimpgVD  39454  hbalgVD  39455  hbexgVD  39456  ax6e2eqVD  39457  e2ebindVD  39462  vk15.4jVD  39464