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Theorem hba1 1978
Description:  x is not free in  A. x ph. 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.)
Assertion
Ref Expression
hba1  |-  ( A. x ph  ->  A. x A. x ph )

Proof of Theorem hba1
StepHypRef Expression
1 alex 1698 . 2  |-  ( A. x ph  <->  -.  E. x  -.  ph )
2 hbe1 1917 . . 3  |-  ( E. x  -.  ph  ->  A. x E. x  -.  ph )
32hbn 1977 . 2  |-  ( -. 
E. x  -.  ph  ->  A. x  -.  E. x  -.  ph )
41, 3hbxfrbi 1694 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442   E.wex 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664
This theorem is referenced by:  nfa1  1979  nfald  2034  axi5r  2423  axial  2424  hbra1OLD  2771  bj-19.41al  31250  bj-modal4e  31308  hbntal  36920  hbimpg  36921  hbimpgVD  37301  hbalgVD  37302  hbexgVD  37303  ax6e2eqVD  37304  e2ebindVD  37309  vk15.4jVD  37311
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