MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dral1ALT Structured version   Visualization version   Unicode version

Theorem dral1ALT 2175
Description: Alternate proof of dral1 2174, shorter but requiring ax-11 1937. (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 2173 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
3 axc11 2163 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
4 axc11r 2040 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
53, 4impbid 195 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
62, 5bitrd 261 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator