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Theorem dral1 1709
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 1697 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 dral1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
32biimpd 143 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  ->  ps ) )
41, 3alimdh 1444 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. x ps )
)
5 ax10o 1694 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
64, 5syld 45 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ps )
)
7 hbae 1697 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
82biimprd 157 . . . 4  |-  ( A. x  x  =  y  ->  ( ps  ->  ph )
)
97, 8alimdh 1444 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. y ph )
)
10 ax10o 1694 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ph  ->  A. x ph )
)
1110alequcoms 1497 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ph )
)
129, 11syld 45 . 2  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ph )
)
136, 12impbid 128 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  drnf1  1712  equveli  1733  a16g  1837
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