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

Theorem hbn1fw 1863
Description: Weak version of ax-10 1888 from which we can prove any ax-10 1888 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 . . . 4  |-  ( A. x ph  ->  A. y A. x ph )
2 hbn1fw.2 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
3 hbn1fw.3 . . . 4  |-  ( A. y ps  ->  A. x A. y ps )
4 hbn1fw.4 . . . 4  |-  ( -. 
ph  ->  A. y  -.  ph )
5 hbn1fw.6 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvalw 1859 . . 3  |-  ( A. x ph  <->  A. y ps )
76notbii 298 . 2  |-  ( -. 
A. x ph  <->  -.  A. y ps )
8 hbn1fw.5 . 2  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
97, 8hbxfrbi 1691 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188   A.wal 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840
This theorem depends on definitions:  df-bi 189  df-ex 1661
This theorem is referenced by:  hbn1w  1864
  Copyright terms: Public domain W3C validator