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Theorem spimw 1656
Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
Hypotheses
Ref Expression
spimw.1  |-  ( -. 
ps  ->  A. x  -.  ps )
spimw.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimw  |-  ( A. x ph  ->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem spimw
StepHypRef Expression
1 ax9v 1645 . 2  |-  -.  A. x  -.  x  =  y
2 spimw.1 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
3 spimw.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3spimfw 1636 . 2  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  ps )
)
51, 4ax-mp 8 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  spimvw  1657  spnfw  1658  cbvaliw  1659  spfw  1676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-9 1644
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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