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Theorem spimw 1852
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 ax6v 1816 . 2  |-  -.  A. x  -.  x  =  y
2 spimw.1 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
3 spimw.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3spimfw 1805 . 2  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  ps )
)
51, 4ax-mp 5 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-6 1815
This theorem depends on definitions:  df-bi 190  df-ex 1674
This theorem is referenced by:  spimvw  1853  spnfw  1854  cbvaliw  1859  spfw  1885
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