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Theorem ceqsalgALT 3135
Description: Alternate proof of ceqsalg 3134, not using ceqsalt 3132. (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 3120 . . 3  |-  ( A  e.  V  ->  E. x  x  =  A )
2 nfa1 1898 . . . 4  |-  F/ x A. x ( x  =  A  ->  ph )
3 ceqsalg.1 . . . 4  |-  F/ x ps
4 ceqsalg.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54biimpd 207 . . . . . 6  |-  ( x  =  A  ->  ( ph  ->  ps ) )
65a2i 13 . . . . 5  |-  ( ( x  =  A  ->  ph )  ->  ( x  =  A  ->  ps ) )
76sps 1866 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) )
82, 3, 7exlimd 1915 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  ps ) )
91, 8syl5com 30 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
104biimprcd 225 . . 3  |-  ( ps 
->  ( x  =  A  ->  ph ) )
113, 10alrimi 1878 . 2  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
129, 11impbid1 203 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    = wceq 1395   E.wex 1613   F/wnf 1617    e. wcel 1819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator