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Theorem iota5f 27403
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 2604 . . . 4  |-  F/ x  A  e.  V
41, 3nfan 1861 . . 3  |-  F/ x
( ph  /\  A  e.  V )
5 iota5f.3 . . 3  |-  ( (
ph  /\  A  e.  V )  ->  ( ps 
<->  x  =  A ) )
64, 5alrimi 1811 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  <->  x  =  A ) )
72nfeq2 2605 . . . . . 6  |-  F/ x  y  =  A
8 eqeq2 2452 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
98bibi2d 318 . . . . . 6  |-  ( y  =  A  ->  (
( ps  <->  x  =  y )  <->  ( ps  <->  x  =  A ) ) )
107, 9albid 1819 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ps  <->  x  =  y )  <->  A. x
( ps  <->  x  =  A ) ) )
11 eqeq2 2452 . . . . 5  |-  ( y  =  A  ->  (
( iota x ps )  =  y  <->  ( iota x ps )  =  A
) )
1210, 11imbi12d 320 . . . 4  |-  ( y  =  A  ->  (
( A. x ( ps  <->  x  =  y
)  ->  ( iota x ps )  =  y )  <->  ( A. x
( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) ) )
13 iotaval 5413 . . . 4  |-  ( A. x ( ps  <->  x  =  y )  ->  ( iota x ps )  =  y )
1412, 13vtoclg 3051 . . 3  |-  ( A  e.  V  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
1514adantl 466 . 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 184    /\ wa 369   A.wal 1367    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2575   iotacio 5400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2742  df-v 2995  df-sbc 3208  df-un 3354  df-sn 3899  df-pr 3901  df-uni 4113  df-iota 5402
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator