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Theorem ax4wK 27807
Description: Weak version of ax-4 1692. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 9 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
ax4wK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax4wK  |-  ( A. x ph  ->  ph )
Distinct variable groups:    x, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem ax4wK
StepHypRef Expression
1 ax4wK.1 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21biimpd 200 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32a4imvK 27805 . . 3  |-  ( A. x ph  ->  ps )
43ax-gen 1536 . 2  |-  A. y
( A. x ph  ->  ps )
5 ax-5 1533 . . 3  |-  ( A. y ( A. x ph  ->  ps )  -> 
( A. y A. x ph  ->  A. y ps ) )
6 ax-17 1628 . . . 4  |-  ( A. x ph  ->  A. y A. x ph )
7 equcomiK 27793 . . . . . 6  |-  ( y  =  x  ->  x  =  y )
81biimprd 216 . . . . . 6  |-  ( x  =  y  ->  ( ps  ->  ph ) )
97, 8syl 17 . . . . 5  |-  ( y  =  x  ->  ( ps  ->  ph ) )
109a4imvK 27805 . . . 4  |-  ( A. y ps  ->  ph )
116, 10imim12i 55 . . 3  |-  ( ( A. y A. x ph  ->  A. y ps )  ->  ( A. x ph  ->  ph ) )
125, 11syl 17 . 2  |-  ( A. y ( A. x ph  ->  ps )  -> 
( A. x ph  ->  ph ) )
134, 12ax-mp 10 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532
This theorem is referenced by:  ax4vK  27808  hba1wK  27817  ax11wK  27824  ax12o10lem7K  27843
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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