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Theorem ceqsalgALT 3049
Description: Alternate proof of ceqsalg 3048, not using ceqsalt 3046. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ceqsalg.1  |-  F/ x ps
ceqsalg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsalgALT  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsalgALT
StepHypRef Expression
1 elisset 3033 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
2 nfa1 1956 . . . 4  |-  F/ x A. x ( x  =  A  ->  ph )
3 ceqsalg.1 . . . 4  |-  F/ x ps
4 ceqsalg.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 210 . . . . . 6  |-  ( x  =  A  ->  ( ph  ->  ps ) )
65a2i 14 . . . . 5  |-  ( ( x  =  A  ->  ph )  ->  ( x  =  A  ->  ps ) )
76sps 1920 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) )
82, 3, 7exlimd 1974 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  ps ) )
91, 8syl5com 31 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
104biimprcd 228 . . 3  |-  ( ps 
->  ( x  =  A  ->  ph ) )
113, 10alrimi 1932 . 2  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
129, 11impbid1 206 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3024
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator