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Theorem speimfw 1635
Description: Specialization, with additional weakening 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-Aug-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 speimfw.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
21eximi 1566 . 2  |-  ( E. x  x  =  y  ->  E. x ( ph  ->  ps ) )
3 df-ex 1532 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
4 19.35 1590 . 2  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
52, 3, 43imtr3i 256 1  |-  ( -. 
A. x  -.  x  =  y  ->  ( A. x ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531
This theorem is referenced by:  spimfw  1636  19.2OLD  1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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