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Theorem iotain 36838
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 2323 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 vex 3034 . . . . 5  |-  y  e. 
_V
32intsn 4262 . . . 4  |-  |^| { y }  =  y
4 nfa1 1999 . . . . . . 7  |-  F/ x A. x ( ph  <->  x  =  y )
5 sp 1957 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  ->  ( ph 
<->  x  =  y ) )
64, 5abbid 2588 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
7 df-sn 3960 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
86, 7syl6eqr 2523 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
98inteqd 4231 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  |^| { y } )
10 iotaval 5564 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
113, 9, 103eqtr4a 2531 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  ( iota x ph )
)
1211exlimiv 1784 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  |^| { x  | 
ph }  =  ( iota x ph )
)
131, 12sylbi 200 1  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671   E!weu 2319   {cab 2457   {csn 3959   |^|cint 4226   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-sbc 3256  df-un 3395  df-in 3397  df-sn 3960  df-pr 3962  df-uni 4191  df-int 4227  df-iota 5553
This theorem is referenced by: (None)
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