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Theorem ceqex 2911
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 1730 . . 3  |-  ( x  =  A  ->  E. x  x  =  A )
2 isset 2805 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
31, 2sylibr 203 . 2  |-  ( x  =  A  ->  A  e.  _V )
4 eqeq2 2305 . . . 4  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
54anbi1d 685 . . . . . 6  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
65exbidv 1616 . . . . 5  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
76bibi2d 309 . . . 4  |-  ( y  =  A  ->  (
( ph  <->  E. x ( x  =  y  /\  ph ) )  <->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) )
84, 7imbi12d 311 . . 3  |-  ( y  =  A  ->  (
( x  =  y  ->  ( ph  <->  E. x
( x  =  y  /\  ph ) ) )  <->  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) ) )
9 19.8a 1730 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  E. x
( x  =  y  /\  ph ) )
109ex 423 . . . 4  |-  ( x  =  y  ->  ( ph  ->  E. x ( x  =  y  /\  ph ) ) )
11 vex 2804 . . . . . 6  |-  y  e. 
_V
1211alexeq 2910 . . . . 5  |-  ( A. x ( x  =  y  ->  ph )  <->  E. x
( x  =  y  /\  ph ) )
13 sp 1728 . . . . . 6  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
1413com12 27 . . . . 5  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
1512, 14syl5bir 209 . . . 4  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
1610, 15impbid 183 . . 3  |-  ( x  =  y  ->  ( ph 
<->  E. x ( x  =  y  /\  ph ) ) )
178, 16vtoclg 2856 . 2  |-  ( A  e.  _V  ->  (
x  =  A  -> 
( ph  <->  E. x ( x  =  A  /\  ph ) ) ) )
183, 17mpcom 32 1  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  ceqsexg  2912  sbc6g  3029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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