Theorem hba1 2330

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 2189	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nf5ri 2234	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  axi5r  2729  axial  2730  bj-19.41al  37340  bj-wnf1  37403  hbntal  45322  hbimpg  45323  hbimpgVD  45672  hbalgVD  45673  hbexgVD  45674  ax6e2eqVD  45675  e2ebindVD  45680  vk15.4jVD  45682  quantgodelALT  47649