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Theorem iotavalsb 36854
Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotavalsb  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x, z)    ps( x, y, z)

Proof of Theorem iotavalsb
StepHypRef Expression
1 19.8a 1955 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) )
2 df-eu 2323 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 iotavalb 36851 . . . 4  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
4 dfsbcq 3257 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
54eqcoms 2479 . . . 4  |-  ( ( iota x ph )  =  y  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
63, 5syl6bi 236 . . 3  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
72, 6sylbir 218 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  ( A. x
( ph  <->  x  =  y
)  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
81, 7mpcom 36 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452   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-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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