Theorem hba1 2136

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 2014	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2052	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033

This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700

This theorem is referenced by:  nfa1OLD  2142  nfaldOLD  2151  nfa1OLDOLD  2194  axi5r  2581  axial  2582  bj-19.41al  31632  bj-modal4e  31698  hbntal  37586  hbimpg  37587  hbimpgVD  37958  hbalgVD  37959  hbexgVD  37960  ax6e2eqVD  37961  e2ebindVD  37966  vk15.4jVD  37968