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Theorem iota5 5555
Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)
Hypothesis
Ref Expression
iota5.1  |-  ( (
ph  /\  A  e.  V )  ->  ( ps 
<->  x  =  A ) )
Assertion
Ref Expression
iota5  |-  ( (
ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
Distinct variable groups:    x, A    x, V    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem iota5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iota5.1 . . 3  |-  ( (
ph  /\  A  e.  V )  ->  ( ps 
<->  x  =  A ) )
21alrimiv 1742 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  <->  x  =  A ) )
3 eqeq2 2419 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
43bibi2d 318 . . . . . 6  |-  ( y  =  A  ->  (
( ps  <->  x  =  y )  <->  ( ps  <->  x  =  A ) ) )
54albidv 1736 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ps  <->  x  =  y )  <->  A. x
( ps  <->  x  =  A ) ) )
6 eqeq2 2419 . . . . 5  |-  ( y  =  A  ->  (
( iota x ps )  =  y  <->  ( iota x ps )  =  A
) )
75, 6imbi12d 320 . . . 4  |-  ( y  =  A  ->  (
( A. x ( ps  <->  x  =  y
)  ->  ( iota x ps )  =  y )  <->  ( A. x
( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) ) )
8 iotaval 5546 . . . 4  |-  ( A. x ( ps  <->  x  =  y )  ->  ( iota x ps )  =  y )
97, 8vtoclg 3119 . . 3  |-  ( A  e.  V  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
109adantl 466 . 2  |-  ( (
ph  /\  A  e.  V )  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
112, 10mpd 15 1  |-  ( (
ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369   A.wal 1405    = wceq 1407    e. wcel 1844   iotacio 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rex 2762  df-v 3063  df-sbc 3280  df-un 3421  df-sn 3975  df-pr 3977  df-uni 4194  df-iota 5535
This theorem is referenced by:  isf32lem9  8775  rlimdm  13525  fsum  13693  fprod  13902  gsumval2a  16232  dchrptlem1  23922  lgsdchrval  24005  rlimdmafv  37643
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