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Theorem iotasbc2 27723
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 27722 . 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 27722 . . . . 5  |-  ( E! x ps  ->  ( [. ( iota x ps )  /  z ]. ch 
<->  E. z ( A. x ( ps  <->  x  =  z )  /\  ch ) ) )
32anbi2d 684 . . . 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 938 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z )  /\  ch ) 
<->  ( A. x (
ph 
<->  x  =  y )  /\  ( A. x
( ps  <->  x  =  z )  /\  ch ) ) )
54exbii 1572 . . . . 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 1858 . . . . 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 241 . . . 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 252 . . 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 1616 . 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 680 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 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632   E!weu 2156   [.wsbc 3004   iotacio 5233
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-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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