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Theorem iota5 5585
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 1767 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  <->  x  =  A ) )
3 eqeq2 2437 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
43bibi2d 319 . . . . . 6  |-  ( y  =  A  ->  (
( ps  <->  x  =  y )  <->  ( ps  <->  x  =  A ) ) )
54albidv 1761 . . . . 5  |-  ( y  =  A  ->  ( A. x ( ps  <->  x  =  y )  <->  A. x
( ps  <->  x  =  A ) ) )
6 eqeq2 2437 . . . . 5  |-  ( y  =  A  ->  (
( iota x ps )  =  y  <->  ( iota x ps )  =  A
) )
75, 6imbi12d 321 . . . 4  |-  ( y  =  A  ->  (
( A. x ( ps  <->  x  =  y
)  ->  ( iota x ps )  =  y )  <->  ( A. x
( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) ) )
8 iotaval 5576 . . . 4  |-  ( A. x ( ps  <->  x  =  y )  ->  ( iota x ps )  =  y )
97, 8vtoclg 3139 . . 3  |-  ( A  e.  V  ->  ( A. x ( ps  <->  x  =  A )  ->  ( iota x ps )  =  A ) )
109adantl 467 . 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 187    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1872   iotacio 5563
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 2057  ax-ext 2401
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 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565
This theorem is referenced by:  isf32lem9  8798  rlimdm  13614  fsum  13785  fprod  13994  gsumval2a  16521  dchrptlem1  24190  lgsdchrval  24273  rlimdmafv  38549
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