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Theorem dral1-o 2230
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2053 using ax-10o 2215. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
Hypothesis
Ref Expression
dral1-o.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dral1-o  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )

Proof of Theorem dral1-o
StepHypRef Expression
1 hbae-o 2229 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 dral1-o.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
32biimpd 199 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  ->  ps ) )
41, 3alimdh 1572 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. x ps )
)
5 ax-10o 2215 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
64, 5syld 42 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ps )
)
7 hbae-o 2229 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
82biimprd 215 . . . 4  |-  ( A. x  x  =  y  ->  ( ps  ->  ph )
)
97, 8alimdh 1572 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. y ph )
)
10 ax-10o 2215 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ph  ->  A. x ph )
)
1110aecoms-o 2228 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ph )
)
129, 11syld 42 . 2  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ph )
)
136, 12impbid 184 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549
This theorem is referenced by:  ax11  2231  a16g-o  2262  ax11indalem  2273  ax11inda2ALT  2274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-7 1749  ax-4 2211  ax-5o 2212  ax-6o 2213  ax-10o 2215  ax-12o 2218
This theorem depends on definitions:  df-bi 178  df-ex 1551
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