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Theorem hba1 2325
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 → ∀𝑥𝑥𝜑)

Proof of Theorem hba1
StepHypRef Expression
1 nfa1 2193 . 2 𝑥𝑥𝜑
21nf5ri 2227 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-or 874  df-ex 1875  df-nf 1879
This theorem is referenced by:  axi5r  2735  axial  2736  bj-19.41al  33094  bj-modal4e  33161  hbntal  39455  hbimpg  39456  hbimpgVD  39816  hbalgVD  39817  hbexgVD  39818  ax6e2eqVD  39819  e2ebindVD  39824  vk15.4jVD  39826
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