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Theorem iotain 36681
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 2275 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 vex 3020 . . . . 5  |-  y  e. 
_V
32intsn 4230 . . . 4  |-  |^| { y }  =  y
4 nfa1 1956 . . . . . . 7  |-  F/ x A. x ( ph  <->  x  =  y )
5 sp 1914 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  ->  ( ph 
<->  x  =  y ) )
64, 5abbid 2540 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
7 df-sn 3937 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
86, 7syl6eqr 2475 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
98inteqd 4198 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  |^| { y } )
10 iotaval 5514 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
113, 9, 103eqtr4a 2483 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  ( iota x ph )
)
1211exlimiv 1770 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  |^| { x  | 
ph }  =  ( iota x ph )
)
131, 12sylbi 198 1  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1657   E!weu 2271   {cab 2409   {csn 3936   |^|cint 4193   iotacio 5501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ral 2714  df-rex 2715  df-v 3019  df-sbc 3238  df-un 3379  df-in 3381  df-sn 3937  df-pr 3939  df-uni 4158  df-int 4194  df-iota 5503
This theorem is referenced by: (None)
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