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Theorem dral1 1659
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 1647 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 dral1.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
32biimpd 142 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  ->  ps ) )
41, 3alimdh 1397 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. x ps )
)
5 ax10o 1644 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
64, 5syld 44 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ps )
)
7 hbae 1647 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
82biimprd 156 . . . 4  |-  ( A. x  x  =  y  ->  ( ps  ->  ph )
)
97, 8alimdh 1397 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. y ph )
)
10 ax10o 1644 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ph  ->  A. x ph )
)
1110alequcoms 1450 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ph )
)
129, 11syld 44 . 2  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ph )
)
136, 12impbid 127 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  drnf1  1662  equveli  1683  a16g  1786
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