Theorem hba1 2306

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 2164	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2209	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-or 855  df-ex 1788  df-nf 1792

This theorem is referenced by:  axi5r  2705  axial  2706  bj-19.41al  37014  bj-wnf1  37077  hbntal  45012  hbimpg  45013  hbimpgVD  45362  hbalgVD  45363  hbexgVD  45364  ax6e2eqVD  45365  e2ebindVD  45370  vk15.4jVD  45372  quantgodelALT  47332