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Theorem dral1ALT 2162
Description: Alternate proof of dral1 2161, shorter but requiring ax-11 1922. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
dral1.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dral1ALT  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )

Proof of Theorem dral1ALT
StepHypRef Expression
1 dral1.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral2 2160 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
3 axc11 2150 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
4 axc112 2022 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
53, 4impbid 194 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
62, 5bitrd 257 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670
This theorem is referenced by: (None)
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