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Theorem hba1wK 28167
Description: Weak version of hba1 1718. See comments for ax6wK 28166. Uses only Tarski's FOL axiom schemes (see description for equidK 28136). (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
ax6wK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hba1wK  |-  ( 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 hba1wK
StepHypRef Expression
1 ax6wK.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21cbvalvK 28163 . . . . . 6  |-  ( A. x ph  <->  A. y ps )
32a1i 12 . . . . 5  |-  ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
43notbid 287 . . . 4  |-  ( x  =  y  ->  ( -.  A. x ph  <->  -.  A. y ps ) )
54ax4wK 28153 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
65con2i 114 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
74ax6wK 28166 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
81ax6wK 28166 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
98con1i 123 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
109alimiK 28142 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
116, 7, 103syl 20 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1532
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-8 1623  ax-17 1628  ax-9v 1632
This theorem depends on definitions:  df-bi 179
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