MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riota5OLD Unicode version

Theorem riota5OLD 6347
Description: A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
riota5OLD.1  |-  ( (
ph  /\  B  e.  A  /\  x  e.  A
)  ->  ( ps  <->  x  =  B ) )
Assertion
Ref Expression
riota5OLD  |-  ( (
ph  /\  B  e.  A )  ->  ( iota_ x  e.  A ps )  =  B )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem riota5OLD
StepHypRef Expression
1 simpr 447 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  e.  A )
2 riota5OLD.1 . . 3  |-  ( (
ph  /\  B  e.  A  /\  x  e.  A
)  ->  ( ps  <->  x  =  B ) )
323expa 1151 . 2  |-  ( ( ( ph  /\  B  e.  A )  /\  x  e.  A )  ->  ( ps 
<->  x  =  B ) )
41, 3riota5 6346 1  |-  ( (
ph  /\  B  e.  A )  ->  ( iota_ x  e.  A ps )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   iota_crio 6313
This theorem is referenced by:  sqr0  11743  lubun  14243  odval2  14882  adjvalval  22533  unxpwdom3  27359  lubunNEW  29785  lub0N  30001  glb0N  30005  trlval2  30974  cdlemefrs32fva  31211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803  df-sbc 3005  df-un 3170  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
  Copyright terms: Public domain W3C validator