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Theorem bj-dral1v 31402
Description: Version of dral1 2169 with a dv condition, which does not require ax-13 2101. Remark: the corresponding versions for dral2 2168 and drex2 2172 are instances of albidv 1777 and exbidv 1778 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-dral1v.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
bj-dral1v  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-dral1v
StepHypRef Expression
1 nfa1 1989 . . 3  |-  F/ x A. x  x  =  y
2 bj-dral1v.1 . . 3  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
31, 2albid 1973 . 2  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. x ps ) )
4 bj-axc11v 31395 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ps 
->  A. y ps )
)
5 axc112 2030 . . 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 1452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by:  bj-drex1v  31403  bj-drnf1v  31404
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