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Theorem iotain 27632
Description: Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
iotain  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)

Proof of Theorem iotain
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-eu 2291 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 vex 2965 . . . . 5  |-  y  e. 
_V
32intsn 4110 . . . 4  |-  |^| { y }  =  y
4 nfa1 1808 . . . . . . 7  |-  F/ x A. x ( ph  <->  x  =  y )
5 sp 1765 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  ->  ( ph 
<->  x  =  y ) )
64, 5abbid 2555 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
7 df-sn 3844 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
86, 7syl6eqr 2492 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
98inteqd 4079 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  |^| { y } )
10 iotaval 5458 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
113, 9, 103eqtr4a 2500 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  ( iota x ph )
)
1211exlimiv 1645 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  |^| { x  | 
ph }  =  ( iota x ph )
)
131, 12sylbi 189 1  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551    = wceq 1653   E!weu 2287   {cab 2428   {csn 3838   |^|cint 4074   iotacio 5445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-v 2964  df-sbc 3168  df-un 3311  df-in 3313  df-sn 3844  df-pr 3845  df-uni 4040  df-int 4075  df-iota 5447
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