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Theorem ax6wK 27816
Description: Weak version of ax-6 1534 from which we can prove any ax-6 1534 instance not involving wff variables or bundling. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
ax6wK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax6wK  |-  ( -. 
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 ax6wK
StepHypRef Expression
1 ax6wK.1 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21cbvalvK 27813 . . . 4  |-  ( A. x ph  <->  A. y ps )
32biimpri 199 . . 3  |-  ( A. y ps  ->  A. x ph )
43con3i 129 . 2  |-  ( -. 
A. x ph  ->  -. 
A. y ps )
5 ax-17 1628 . 2  |-  ( -. 
A. y ps  ->  A. x  -.  A. y ps )
62biimpi 188 . . . 4  |-  ( A. x ph  ->  A. y ps )
76con3i 129 . . 3  |-  ( -. 
A. y ps  ->  -. 
A. x ph )
87alimi 1546 . 2  |-  ( A. x  -.  A. y ps 
->  A. x  -.  A. x ph )
94, 5, 83syl 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 is referenced by:  hba1wK  27817  hbe1wK  27818  ax12o10lem5K  27839  ax12o10lem6K  27841  ax12o10lem7K  27843  ax12o10lem9K  27847
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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