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Theorem hbn1w 1680
Description: Weak version of hbn1 1704. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hbn1w  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
Distinct variable groups:    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem hbn1w
StepHypRef Expression
1 ax-17 1603 . 2  |-  ( A. x ph  ->  A. y A. x ph )
2 ax-17 1603 . 2  |-  ( -. 
ps  ->  A. x  -.  ps )
3 ax-17 1603 . 2  |-  ( A. y ps  ->  A. x A. y ps )
4 ax-17 1603 . 2  |-  ( -. 
ph  ->  A. y  -.  ph )
5 ax-17 1603 . 2  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
6 hbn1w.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
71, 2, 3, 4, 5, 6hbn1fw 1679 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:  hba1w  1681  hbe1w  1682  ax6w  1691
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
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