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Theorem iotain 27617
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 2147 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 vex 2791 . . . . 5  |-  y  e. 
_V
32intsn 3898 . . . 4  |-  |^| { y }  =  y
4 nfa1 1756 . . . . . . 7  |-  F/ x A. x ( ph  <->  x  =  y )
5 sp 1716 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  ->  ( ph 
<->  x  =  y ) )
64, 5abbid 2396 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
x  |  x  =  y } )
7 df-sn 3646 . . . . . 6  |-  { y }  =  { x  |  x  =  y }
86, 7syl6eqr 2333 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  { x  |  ph }  =  {
y } )
98inteqd 3867 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  |^| { y } )
10 iotaval 5230 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
113, 9, 103eqtr4a 2341 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  |^| { x  |  ph }  =  ( iota x ph )
)
1211exlimiv 1666 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  |^| { x  | 
ph }  =  ( iota x ph )
)
131, 12sylbi 187 1  |-  ( E! x ph  ->  |^| { x  |  ph }  =  ( iota x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143   {cab 2269   {csn 3640   |^|cint 3862   iotacio 5217
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
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-in 3159  df-sn 3646  df-pr 3647  df-uni 3828  df-int 3863  df-iota 5219
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