Theorem hba1 2137

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 2015	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2053	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  nfa1OLD  2143  nfaldOLD  2152  nfa1OLDOLD  2195  axi5r  2582  axial  2583  bj-19.41al  31826  bj-modal4e  31892  hbntal  37790  hbimpg  37791  hbimpgVD  38162  hbalgVD  38163  hbexgVD  38164  ax6e2eqVD  38165  e2ebindVD  38170  vk15.4jVD  38172