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Theorem dral1 2058
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
Hypothesis
Ref Expression
dral1.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dral1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )

Proof of Theorem dral1
StepHypRef Expression
1 dral1.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral2 2056 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
3 ax10o 2039 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
4 ax10o2 2025 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
53, 4impbid 185 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
62, 5bitrd 246 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  drex1  2060  drnf1  2062  equveliOLD  2087  a16gALT  2158  sb9i  2172  ralcom2  2874  axpownd  8478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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