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Theorem alexeqd 25065
Description: Two ways to express substitution of  A for  x in  ph. (Contributed by FL, 4-Jun-2012.)
Assertion
Ref Expression
alexeqd  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem alexeqd
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 eqeq2 2305 . . . . . 6  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
x  =  A  <->  x  =  if ( A  e.  _V ,  A ,  (/) ) ) )
32imbi1d 308 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( x  =  A  ->  ph )  <->  ( x  =  if ( A  e. 
_V ,  A ,  (/) )  ->  ph ) ) )
43albidv 1615 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( A. x ( x  =  A  ->  ph )  <->  A. x
( x  =  if ( A  e.  _V ,  A ,  (/) )  ->  ph ) ) )
52anbi1d 685 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( x  =  A  /\  ph )  <->  ( x  =  if ( A  e. 
_V ,  A ,  (/) )  /\  ph )
) )
65exbidv 1616 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( E. x ( x  =  A  /\  ph )  <->  E. x ( x  =  if ( A  e. 
_V ,  A ,  (/) )  /\  ph )
) )
74, 6bibi12d 312 . . 3  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( A. x ( x  =  A  ->  ph )  <->  E. x ( x  =  A  /\  ph ) )  <->  ( A. x ( x  =  if ( A  e. 
_V ,  A ,  (/) )  ->  ph )  <->  E. x
( x  =  if ( A  e.  _V ,  A ,  (/) )  /\  ph ) ) ) )
8 0ex 4166 . . . . 5  |-  (/)  e.  _V
98elimel 3630 . . . 4  |-  if ( A  e.  _V ,  A ,  (/) )  e. 
_V
109alexeq 2910 . . 3  |-  ( A. x ( x  =  if ( A  e. 
_V ,  A ,  (/) )  ->  ph )  <->  E. x
( x  =  if ( A  e.  _V ,  A ,  (/) )  /\  ph ) )
117, 10dedth 3619 . 2  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) ) )
121, 11syl 15 1  |-  ( A  e.  V  ->  ( A. x ( 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   (/)c0 3468   ifcif 3578
This theorem is referenced by:  intopcoaconb  25643
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  ax-nul 4165
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-ne 2461  df-v 2803  df-dif 3168  df-nul 3469  df-if 3579
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