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Theorem bj-equsalv 31411
Description: Version of equsal 2141 with a dv condition, which does not require ax-13 2104. See equsalvw 1855 for a version with two dv conditions requiring fewer axioms. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-equsalv.nf  |-  F/ x ps
bj-equsalv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bj-equsalv  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-equsalv
StepHypRef Expression
1 bj-equsalv.nf . . 3  |-  F/ x ps
2119.23 2013 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 bj-equsalv.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 253 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1699 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 ax6ev 1815 . . 3  |-  E. x  x  =  y
76a1bi 344 . 2  |-  ( ps  <->  ( E. x  x  =  y  ->  ps )
)
82, 5, 73bitr4i 285 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450   E.wex 1671   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-ex 1672  df-nf 1676
This theorem is referenced by:  bj-equsalhv  31412
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