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Theorem iota5f 30309
Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
Hypotheses
Ref Expression
iota5f.1  |-  F/ x ph
iota5f.2  |-  F/_ x A
iota5f.3  |-  ( (
ph  /\  A  e.  V )  ->  ( ps 
<->  x  =  A ) )
Assertion
Ref Expression
iota5f  |-  ( (
ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
Distinct variable group:    x, V
Allowed substitution hints:    ph( x)    ps( x)    A( x)

Proof of Theorem iota5f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iota5f.1 . . . 4  |-  F/ x ph
2 iota5f.2 . . . . 5  |-  F/_ x A
32nfel1 2583 . . . 4  |-  F/ x  A  e.  V
41, 3nfan 1988 . . 3  |-  F/ x
( ph  /\  A  e.  V )
5 iota5f.3 . . 3  |-  ( (
ph  /\  A  e.  V )  ->  ( ps 
<->  x  =  A ) )
64, 5alrimi 1932 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  <->  x  =  A ) )
72nfeq2 2584 . . . . . 6  |-  F/ x  y  =  A
8 eqeq2 2439 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
98bibi2d 319 . . . . . 6  |-  ( y  =  A  ->  (
( ps  <->  x  =  y )  <->  ( ps  <->  x  =  A ) ) )
107, 9albid 1940 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ps  <->  x  =  y )  <->  A. x
( ps  <->  x  =  A ) ) )
11 eqeq2 2439 . . . . 5  |-  ( y  =  A  ->  (
( iota x ps )  =  y  <->  ( iota x ps )  =  A
) )
1210, 11imbi12d 321 . . . 4  |-  ( y  =  A  ->  (
( A. x ( ps  <->  x  =  y
)  ->  ( iota x ps )  =  y )  <->  ( A. x
( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) ) )
13 iotaval 5519 . . . 4  |-  ( A. x ( ps  <->  x  =  y )  ->  ( iota x ps )  =  y )
1412, 13vtoclg 3082 . . 3  |-  ( A  e.  V  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
1514adantl 467 . 2  |-  ( (
ph  /\  A  e.  V )  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
166, 15mpd 15 1  |-  ( (
ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1661    e. wcel 1872   F/_wnfc 2556   iotacio 5506
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 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-v 3024  df-sbc 3243  df-un 3384  df-sn 3942  df-pr 3944  df-uni 4163  df-iota 5508
This theorem is referenced by: (None)
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