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Theorem iotavalsb 27602
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 1762 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) )
2 df-eu 2285 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 iotavalb 27599 . . . 4  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
4 dfsbcq 3156 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
54eqcoms 2439 . . . 4  |-  ( ( iota x ph )  =  y  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
63, 5syl6bi 220 . . 3  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
72, 6sylbir 205 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  ( A. x
( ph  <->  x  =  y
)  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
81, 7mpcom 34 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652   E!weu 2281   [.wsbc 3154   iotacio 5409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2704  df-v 2951  df-sbc 3155  df-un 3318  df-sn 3813  df-pr 3814  df-uni 4009  df-iota 5411
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