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Theorem spimfw 1709
Description: Specialization, with additional weakening (compared to sp 1808) to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
Hypotheses
Ref Expression
spimfw.1  |-  ( -. 
ps  ->  A. x  -.  ps )
spimfw.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimfw  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  ps )
)

Proof of Theorem spimfw
StepHypRef Expression
1 spimfw.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
21speimfw 1707 . 2  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps ) )
3 df-ex 1597 . . 3  |-  ( E. x ps  <->  -.  A. x  -.  ps )
4 spimfw.1 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
54con1i 129 . . 3  |-  ( -. 
A. x  -.  ps  ->  ps )
63, 5sylbi 195 . 2  |-  ( E. x ps  ->  ps )
72, 6syl6 33 1  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1377   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  spimw  1732
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