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Theorem ax4wK 28132
Description: Weak version of ax-4 1692. Uses only Tarski's FOL axiom schemes (see description for equidK 28115). 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 28130 . . 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 28116 . . . . . 6  |-  ( y  =  x  ->  x  =  y )
81biimprd 216 . . . . . 6  |-  ( x  =  y  ->  ( ps  ->  ph ) )
97, 8syl 17 . . . . 5  |-  ( y  =  x  ->  ( ps  ->  ph ) )
109a4imvK 28130 . . . 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  28133  ax4falK  28136  hba1wK  28146  ax11wK  28153  ax12o10lem7K  28172
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
  Copyright terms: Public domain W3C validator