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Theorem bj-equsalhv 31412
Description: Version of equsalh 2142 with a dv condition, which does not require ax-13 2104. Remark: this is the same as equsalhw 2047.

Remarks: equsexvw 1856 has been moved to Main; the theorem axc9lem2 2146 has a dv version which is a simple consequence of ax5e 1768; the theorems nfeqf2 2148, dveeq2 2149, nfeqf1 2150, dveeq1 2151, nfeqf 2152, axc9 2154, ax13 2155, have dv versions which are simple consequences of ax-5 1766. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Hypotheses
Ref Expression
bj-equsalhv.nf  |-  ( ps 
->  A. x ps )
bj-equsalhv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bj-equsalhv  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-equsalhv
StepHypRef Expression
1 bj-equsalhv.nf . . 3  |-  ( ps 
->  A. x ps )
21nfi 1682 . 2  |-  F/ x ps
3 bj-equsalhv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3bj-equsalv 31411 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450
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: (None)
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