MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hba1 Unicode version

Theorem hba1 1718
Description:  x is not free in  A. x ph. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hba1  |-  ( A. x ph  ->  A. x A. x ph )

Proof of Theorem hba1
StepHypRef Expression
1 ax-4 1692 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
21con2i 114 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
3 ax-6 1534 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
4 ax-6 1534 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
54con1i 123 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
65alimi 1546 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
72, 3, 63syl 20 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  nfa1  1719  ax67to6  1750  ax467to6  1754  hbim1  1810  dvelimfALT  1854  dvelimf-o  1855  ax11indalem  2113  ax11inda2ALT  2114  ax11inda  2116  hbra1  2565  hbntal  27356  hbimpg  27357  hbimpgVD  27714  hbalgVD  27715  hbexgVD  27716  a9e2eqVD  27717  e2ebindVD  27722  vk15.4jVD  27724  ax12-2  28254  ax12OLD  28256  a12studyALT  28284  a12study3  28286
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-4 1692
  Copyright terms: Public domain W3C validator