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Theorem dral1-o 32387
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 2120 using ax-c11 32372. (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 32386 . . . 4  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
2 dral1-o.1 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
32biimpd 210 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  ->  ps ) )
41, 3alimdh 1685 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. x ps )
)
5 ax-c11 32372 . . 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-o 32386 . . . 4  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
82biimprd 226 . . . 4  |-  ( A. x  x  =  y  ->  ( ps  ->  ph )
)
97, 8alimdh 1685 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. y ph )
)
10 ax-c11 32372 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ph  ->  A. x ph )
)
1110aecoms-o 32385 . . 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 193 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-11 1891  ax-c5 32368  ax-c4 32369  ax-c7 32370  ax-c11 32372  ax-c9 32375
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  ax12  32388  axc16g-o  32418  ax12indalem  32429  ax12inda2ALT  32430
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