MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbn1fw Structured version   Unicode version

Theorem hbn1fw 1721
Description: Weak version of ax-6 1746 from which we can prove any ax-6 1746 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
Hypotheses
Ref Expression
hbn1fw.1  |-  ( A. x ph  ->  A. y A. x ph )
hbn1fw.2  |-  ( -. 
ps  ->  A. x  -.  ps )
hbn1fw.3  |-  ( A. y ps  ->  A. x A. y ps )
hbn1fw.4  |-  ( -. 
ph  ->  A. y  -.  ph )
hbn1fw.5  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
hbn1fw.6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hbn1fw  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem hbn1fw
StepHypRef Expression
1 hbn1fw.1 . . . . 5  |-  ( A. x ph  ->  A. y A. x ph )
2 hbn1fw.2 . . . . 5  |-  ( -. 
ps  ->  A. x  -.  ps )
3 hbn1fw.3 . . . . 5  |-  ( A. y ps  ->  A. x A. y ps )
4 hbn1fw.4 . . . . 5  |-  ( -. 
ph  ->  A. y  -.  ph )
5 hbn1fw.6 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvalw 1716 . . . 4  |-  ( A. x ph  <->  A. y ps )
76notbii 289 . . 3  |-  ( -. 
A. x ph  <->  -.  A. y ps )
87biimpi 188 . 2  |-  ( -. 
A. x ph  ->  -. 
A. y ps )
9 hbn1fw.5 . 2  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
107biimpri 199 . . 3  |-  ( -. 
A. y ps  ->  -. 
A. x ph )
1110alimi 1569 . 2  |-  ( A. x  -.  A. y ps 
->  A. x  -.  A. x ph )
128, 9, 113syl 19 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  hbn1w  1723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
  Copyright terms: Public domain W3C validator