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Theorem equsalhw 2038
Description: Weaker version of equsalh 2139 (requiring distinct variables) without using ax-13 2101. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
Hypotheses
Ref Expression
equsalhw.1  |-  ( ps 
->  A. x ps )
equsalhw.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsalhw  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsalhw
StepHypRef Expression
1 equsalhw.1 . . 3  |-  ( ps 
->  A. x ps )
2119.23h 2004 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 equsalhw.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 253 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1701 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 ax6ev 1817 . . 3  |-  E. x  x  =  y
76a1bi 343 . 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 1452   E.wex 1673
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:  dvelimhw  2070
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