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Theorem dral1 1918
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.)
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 hbae 1906 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 dral1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
32biimpd 198 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  ->  ps ) )
41, 3alimdh 1553 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. x ps )
)
5 ax10o 1905 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
64, 5syld 40 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ps )
)
7 hbae 1906 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
82biimprd 214 . . . 4  |-  ( A. x  x  =  y  ->  ( ps  ->  ph )
)
97, 8alimdh 1553 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. y ph )
)
10 ax10o 1905 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ph  ->  A. x ph )
)
1110aecoms 1900 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ph )
)
129, 11syld 40 . 2  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ph )
)
136, 12impbid 183 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  drex1  1920  drnf1  1922  equveli  1941  a16gALT  2002  sb9i  2047  ralcom2  2717  axpownd  8239  ax12-2  29725  ax12-4  29728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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