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Theorem bj-drnf1v 32275
Description: Version of drnf1 2021 with a dv condition, which does not require ax-13 1943. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-drnf1v.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
bj-drnf1v  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-drnf1v
StepHypRef Expression
1 bj-drnf1v.1 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
21bj-dral1v 32273 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ps ) )
31, 2imbi12d 320 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ph )  <->  ( ps  ->  A. y ps )
) )
43bj-dral1v 32273 . 2  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  A. x ph )  <->  A. y ( ps  ->  A. y ps ) ) )
5 df-nf 1590 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
6 df-nf 1590 . 2  |-  ( F/ y ps  <->  A. y
( ps  ->  A. y ps ) )
74, 5, 63bitr4g 288 1  |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1367   F/wnf 1589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-ex 1587  df-nf 1590
This theorem is referenced by: (None)
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