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Theorem hba1 2297
 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 2151 . 2 𝑥𝑥𝜑
21nf5ri 2191 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173 This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781 This theorem is referenced by:  axi5r  2783  axial  2784  bj-19.41al  33987  bj-wnf1  34046  hbntal  40880  hbimpg  40881  hbimpgVD  41231  hbalgVD  41232  hbexgVD  41233  ax6e2eqVD  41234  e2ebindVD  41239  vk15.4jVD  41241
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