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Theorem iotasbc2 36841
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 36840 . 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 36840 . . . . 5  |-  ( E! x ps  ->  ( [. ( iota x ps )  /  z ]. ch 
<->  E. z ( A. x ( ps  <->  x  =  z )  /\  ch ) ) )
32anbi2d 718 . . . 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 1011 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z )  /\  ch ) 
<->  ( A. x (
ph 
<->  x  =  y )  /\  ( A. x
( ps  <->  x  =  z )  /\  ch ) ) )
54exbii 1726 . . . . 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 1842 . . . . 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 258 . . . 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 269 . . 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 1776 . 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 714 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   A.wal 1450   E.wex 1671   E!weu 2319   [.wsbc 3255   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sbc 3256  df-un 3395  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553
This theorem is referenced by: (None)
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