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Theorem iotavalsb 31294
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 1843 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) )
2 df-eu 2272 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 iotavalb 31291 . . . 4  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
4 dfsbcq 3315 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
54eqcoms 2455 . . . 4  |-  ( ( iota x ph )  =  y  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
63, 5syl6bi 228 . . 3  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
72, 6sylbir 213 . 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 184   A.wal 1381    = wceq 1383   E.wex 1599   E!weu 2268   [.wsbc 3313   iotacio 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-v 3097  df-sbc 3314  df-un 3466  df-sn 4015  df-pr 4017  df-uni 4235  df-iota 5541
This theorem is referenced by: (None)
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