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Theorem dral1 2170
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 1931. (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 1990 . . 3  |-  F/ x A. x  x  =  y
2 dral1.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
31, 2albid 1974 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
4 axc11 2159 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
5 axc112 2031 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ps 
->  A. x ps )
)
64, 5impbid 195 . 2  |-  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ps ) )
73, 6bitrd 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 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  drex1  2172  drnf1  2174  axc16gALT  2207  sb9  2266  ralcom2  2967  axpownd  9057  wl-dral1d  31910  wl-ax12  31959  wl-ax11-lem5  31965  wl-ax11-lem8  31968  wl-ax11-lem9  31969
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