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Theorem ceqex 3011
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
Assertion
Ref Expression
ceqex  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ceqex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 19.8a 1754 . . 3  |-  ( x  =  A  ->  E. x  x  =  A )
2 isset 2905 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
31, 2sylibr 204 . 2  |-  ( x  =  A  ->  A  e.  _V )
4 eqeq2 2398 . . . 4  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
54anbi1d 686 . . . . . 6  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
65exbidv 1633 . . . . 5  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
76bibi2d 310 . . . 4  |-  ( y  =  A  ->  (
( ph  <->  E. x ( x  =  y  /\  ph ) )  <->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) )
84, 7imbi12d 312 . . 3  |-  ( y  =  A  ->  (
( x  =  y  ->  ( ph  <->  E. x
( x  =  y  /\  ph ) ) )  <->  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) ) )
9 19.8a 1754 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  E. x
( x  =  y  /\  ph ) )
109ex 424 . . . 4  |-  ( x  =  y  ->  ( ph  ->  E. x ( x  =  y  /\  ph ) ) )
11 vex 2904 . . . . . 6  |-  y  e. 
_V
1211alexeq 3010 . . . . 5  |-  ( A. x ( x  =  y  ->  ph )  <->  E. x
( x  =  y  /\  ph ) )
13 sp 1755 . . . . . 6  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
1413com12 29 . . . . 5  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
1512, 14syl5bir 210 . . . 4  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
1610, 15impbid 184 . . 3  |-  ( x  =  y  ->  ( ph 
<->  E. x ( x  =  y  /\  ph ) ) )
178, 16vtoclg 2956 . 2  |-  ( A  e.  _V  ->  (
x  =  A  -> 
( ph  <->  E. x ( x  =  A  /\  ph ) ) ) )
183, 17mpcom 34 1  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2901
This theorem is referenced by:  ceqsexg  3012  sbc6g  3131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903
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