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

Step	Hyp	Ref	Expression
1		alex 1698	. 2 |- ( A. x ph <-> -. E. x -. ph )
2		hbe1 1917	. . 3 |- ( E. x -. ph -> A. x E. x -. ph )
3	2	hbn 1977	. 2 |- ( -. E. x -. ph -> A. x -. E. x -. ph )
4	1, 3	hbxfrbi 1694	1 |- ( A. x ph -> A. x A. x ph )

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