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Theorem speimfw 1785
Description: Specialization, with additional weakening (compared to 19.2 1801) 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, 5-Dec-2017.)
Hypothesis
Ref Expression
speimfw.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
speimfw  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps ) )

Proof of Theorem speimfw
StepHypRef Expression
1 df-ex 1660 . . 3  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
21biimpri 209 . 2  |-  ( -. 
A. x  -.  x  =  y  ->  E. x  x  =  y )
3 speimfw.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43com12 32 . . 3  |-  ( ph  ->  ( x  =  y  ->  ps ) )
54aleximi 1700 . 2  |-  ( A. x ph  ->  ( E. x  x  =  y  ->  E. x ps )
)
62, 5syl5com 31 1  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  spimfw  1787
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