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Theorem dral1 2027
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.) Remove dependency on ax-11 1782. (Revised by Wolf Lammen, 6-Sep-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 nfa1 1836 . . 3  |-  F/ x A. x  x  =  y
2 dral1.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
31, 2albid 1824 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
4 axc11 2014 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
5 axc112 1875 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
64, 5impbid 191 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
73, 6bitrd 253 1  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  drex1  2029  drnf1  2031  axc16gALT  2066  sb9  2136  sb9iOLD  2138  ralcom2  2991  axpownd  8881  wl-dral1d  28528  wl-ax11-lem5  28573  wl-ax11-lem8  28576  wl-ax11-lem9  28577
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