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Theorem a4imvK 28151
Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes (see description for equidK 28136). (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
a4imvK.1  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
a4imvK  |-  ( A. x ph  ->  ps )
Distinct variable groups:    x, y    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem a4imvK
StepHypRef Expression
1 ax-9v 1632 . 2  |-  -.  A. x  -.  x  =  y
2 ax-17 1628 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
3 a4imvK.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43com12 29 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ps ) )
54con3rr3 130 . . . . 5  |-  ( -. 
ps  ->  ( ph  ->  -.  x  =  y ) )
65alimiK 28142 . . . 4  |-  ( A. x  -.  ps  ->  A. x
( ph  ->  -.  x  =  y ) )
7 ax-5 1533 . . . 4  |-  ( A. x ( ph  ->  -.  x  =  y )  ->  ( A. x ph  ->  A. x  -.  x  =  y ) )
82, 6, 73syl 20 . . 3  |-  ( -. 
ps  ->  ( A. x ph  ->  A. x  -.  x  =  y ) )
98com12 29 . 2  |-  ( A. x ph  ->  ( -.  ps  ->  A. x  -.  x  =  y ) )
101, 9mt3i 120 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   A.wal 1532
This theorem is referenced by:  ax4wK  28153  cbvalivK  28161  ax7wK  28169
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-17 1628  ax-9v 1632
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