Theorem hba1 2327

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 2185	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2230	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by:  axi5r  2726  axial  2727  bj-19.41al  37131  bj-wnf1  37194  hbntal  45129  hbimpg  45130  hbimpgVD  45479  hbalgVD  45480  hbexgVD  45481  ax6e2eqVD  45482  e2ebindVD  45487  vk15.4jVD  45489  quantgodelALT  47449