Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iotasbc2 Unicode version

Theorem iotasbc2 26953
Description: Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotasbc2  |-  ( ( E! x ph  /\  E! x ps )  -> 
( [. ( iota x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
( A. x (
ph 
<->  x  =  y )  /\  A. x ( ps  <->  x  =  z
)  /\  ch )
) )
Distinct variable groups:    x, y,
z    ph, y, z    ps, y, z
Allowed substitution hints:    ph( x)    ps( x)    ch( x, y, z)

Proof of Theorem iotasbc2
StepHypRef Expression
1 iotasbc 26952 . 2  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. [. ( iota x ps )  / 
z ]. ch  <->  E. y
( A. x (
ph 
<->  x  =  y )  /\  [. ( iota
x ps )  / 
z ]. ch ) ) )
2 iotasbc 26952 . . . . 5  |-  ( E! x ps  ->  ( [. ( iota x ps )  /  z ]. ch 
<->  E. z ( A. x ( ps  <->  x  =  z )  /\  ch ) ) )
32anbi2d 687 . . . 4  |-  ( E! x ps  ->  (
( A. x (
ph 
<->  x  =  y )  /\  [. ( iota
x ps )  / 
z ]. ch )  <->  ( A. x ( ph  <->  x  =  y )  /\  E. z ( A. x
( ps  <->  x  =  z )  /\  ch ) ) ) )
4 3anass 943 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z )  /\  ch ) 
<->  ( A. x (
ph 
<->  x  =  y )  /\  ( A. x
( ps  <->  x  =  z )  /\  ch ) ) )
54exbii 1580 . . . . 5  |-  ( E. z ( A. x
( ph  <->  x  =  y
)  /\  A. x
( ps  <->  x  =  z )  /\  ch ) 
<->  E. z ( A. x ( ph  <->  x  =  y )  /\  ( A. x ( ps  <->  x  =  z )  /\  ch ) ) )
6 19.42v 2039 . . . . 5  |-  ( E. z ( A. x
( ph  <->  x  =  y
)  /\  ( A. x ( ps  <->  x  =  z )  /\  ch ) )  <->  ( A. x ( ph  <->  x  =  y )  /\  E. z ( A. x
( ps  <->  x  =  z )  /\  ch ) ) )
75, 6bitr2i 243 . . . 4  |-  ( ( A. x ( ph  <->  x  =  y )  /\  E. z ( A. x
( ps  <->  x  =  z )  /\  ch ) )  <->  E. z
( A. x (
ph 
<->  x  =  y )  /\  A. x ( ps  <->  x  =  z
)  /\  ch )
)
83, 7syl6bb 254 . . 3  |-  ( E! x ps  ->  (
( A. x (
ph 
<->  x  =  y )  /\  [. ( iota
x ps )  / 
z ]. ch )  <->  E. z
( A. x (
ph 
<->  x  =  y )  /\  A. x ( ps  <->  x  =  z
)  /\  ch )
) )
98exbidv 2006 . 2  |-  ( E! x ps  ->  ( E. y ( A. x
( ph  <->  x  =  y
)  /\  [. ( iota
x ps )  / 
z ]. ch )  <->  E. y E. z ( A. x
( ph  <->  x  =  y
)  /\  A. x
( ps  <->  x  =  z )  /\  ch ) ) )
101, 9sylan9bb 683 1  |-  ( ( E! x ph  /\  E! x ps )  -> 
( [. ( iota x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
( A. x (
ph 
<->  x  =  y )  /\  A. x ( ps  <->  x  =  z
)  /\  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939   A.wal 1532   E.wex 1537    = wceq 1619   E!weu 2117   [.wsbc 2935   iotacio 6188
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-v 2742  df-sbc 2936  df-un 3099  df-sn 3587  df-pr 3588  df-uni 3769  df-iota 6190
  Copyright terms: Public domain W3C validator