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Theorem iota5 4968
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 1799 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  <->  x  =  A ) )
3 eqeq2 2094 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
43bibi2d 230 . . . . . 6  |-  ( y  =  A  ->  (
( ps  <->  x  =  y )  <->  ( ps  <->  x  =  A ) ) )
54albidv 1749 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ps  <->  x  =  y )  <->  A. x
( ps  <->  x  =  A ) ) )
6 eqeq2 2094 . . . . 5  |-  ( y  =  A  ->  (
( iota x ps )  =  y  <->  ( iota x ps )  =  A
) )
75, 6imbi12d 232 . . . 4  |-  ( y  =  A  ->  (
( A. x ( ps  <->  x  =  y
)  ->  ( iota x ps )  =  y )  <->  ( A. x
( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) ) )
8 iotaval 4959 . . . 4  |-  ( A. x ( ps  <->  x  =  y )  ->  ( iota x ps )  =  y )
97, 8vtoclg 2672 . . 3  |-  ( A  e.  V  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
109adantl 271 . 2  |-  ( (
ph  /\  A  e.  V )  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
112, 10mpd 13 1  |-  ( (
ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1285    = wceq 1287    e. wcel 1436   iotacio 4946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-sn 3437  df-pr 3438  df-uni 3639  df-iota 4948
This theorem is referenced by:  fisum  10670
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