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Theorem hbn1fw 1679
Description: Weak version of ax-6 1703 from which we can prove any ax-6 1703 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
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 1675 . . . 4  |-  ( A. x ph  <->  A. y ps )
76biimpri 197 . . 3  |-  ( A. y ps  ->  A. x ph )
87con3i 127 . 2  |-  ( -. 
A. x ph  ->  -. 
A. y ps )
9 hbn1fw.5 . 2  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
106biimpi 186 . . . 4  |-  ( A. x ph  ->  A. y ps )
1110con3i 127 . . 3  |-  ( -. 
A. y ps  ->  -. 
A. x ph )
1211alimi 1546 . 2  |-  ( A. x  -.  A. y ps 
->  A. x  -.  A. x ph )
138, 9, 123syl 18 1  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  hbn1w  1680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator