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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 → ∀𝑥𝑥𝜑)

Proof of Theorem hba1
StepHypRef Expression
1 nfa1 2014 . 2 𝑥𝑥𝜑
21nf5ri 2052 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
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
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