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Theorem iotasbc2 27488
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 27487 . 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 27487 . . . . 5  |-  ( E! x ps  ->  ( [. ( iota x ps )  /  z ]. ch 
<->  E. z ( A. x ( ps  <->  x  =  z )  /\  ch ) ) )
32anbi2d 685 . . . 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 940 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z )  /\  ch ) 
<->  ( A. x (
ph 
<->  x  =  y )  /\  ( A. x
( ps  <->  x  =  z )  /\  ch ) ) )
54exbii 1589 . . . . 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 1924 . . . . 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 242 . . . 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 253 . . 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 1633 . 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 681 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 177    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547   E!weu 2254   [.wsbc 3121   iotacio 5375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-sbc 3122  df-un 3285  df-sn 3780  df-pr 3781  df-uni 3976  df-iota 5377
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