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Theorem dral2 2023
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, 27-Feb-2005.) Revised to allow a shortening of dral1 2024. (Revised by Wolf Lammen, 4-Mar-2018.)
Hypothesis
Ref Expression
dral1.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dral2  |-  ( A. x  x  =  y  ->  ( A. z ph  <->  A. z ps ) )

Proof of Theorem dral2
StepHypRef Expression
1 nfae 2013 . 2  |-  F/ z A. x  x  =  y
2 dral1.1 . 2  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
31, 2albid 1821 1  |-  ( A. x  x  =  y  ->  ( A. z ph  <->  A. z 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-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  dral1ALT  2025  sbal1  2178  sbal1OLD  2179  sbal2  2180  drnfc1  2631  drnfc2  2632  axpownd  8871  wl-sbalnae  28529
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