MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dral1 Unicode version

Theorem dral1 2022
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.) (Proof shortened by Wolf Lammen, 4-Mar-2018.)
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 dral1.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21dral2 2020 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
3 ax10o 2001 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
4 ax10 1991 . . . 4  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
5 ax10o 2001 . . . 4  |-  ( A. y  y  =  x  ->  ( A. y ps 
->  A. x ps )
)
64, 5syl 16 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
73, 6impbid 184 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
82, 7bitrd 245 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 1546
This theorem is referenced by:  drex1  2024  drnf1  2026  equveliOLD  2043  a16gALT  2098  sb9i  2143  ralcom2  2832  axpownd  8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator