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Theorem iotain 31565
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 2288 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 vex 3109 . . . . 5  |-  y  e. 
_V
32intsn 4308 . . . 4  |-  |^| { y }  =  y
4 nfa1 1902 . . . . . . 7  |-  F/ x A. x ( ph  <->  x  =  y )
5 sp 1864 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  ->  ( ph 
<->  x  =  y ) )
64, 5abbid 2589 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
7 df-sn 4017 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
86, 7syl6eqr 2513 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
98inteqd 4276 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  |^| { y } )
10 iotaval 5545 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
113, 9, 103eqtr4a 2521 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  ( iota x ph )
)
1211exlimiv 1727 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  |^| { x  | 
ph }  =  ( iota x ph )
)
131, 12sylbi 195 1  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398   E.wex 1617   E!weu 2284   {cab 2439   {csn 4016   |^|cint 4271   iotacio 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-un 3466  df-in 3468  df-sn 4017  df-pr 4019  df-uni 4236  df-int 4272  df-iota 5534
This theorem is referenced by: (None)
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