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Theorem alexeqd 24962
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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 eqeq2 2292 . . . . . 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 1611 . . . 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 1612 . . . 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 4150 . . . . 5  |-  (/)  e.  _V
98elimel 3617 . . . 4  |-  if ( A  e.  _V ,  A ,  (/) )  e. 
_V
109alexeq 2897 . . 3  |-  ( A. x ( x  =  if ( A  e. 
_V ,  A ,  (/) )  ->  ph )  <->  E. x
( x  =  if ( A  e.  _V ,  A ,  (/) )  /\  ph ) )
117, 10dedth 3606 . 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ifcif 3565
This theorem is referenced by:  intopcoaconb  25540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-nul 3456  df-if 3566
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