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Theorem iotavalsb 36642
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 1908 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) )
2 df-eu 2269 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 iotavalb 36639 . . . 4  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
4 dfsbcq 3301 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
54eqcoms 2434 . . . 4  |-  ( ( iota x ph )  =  y  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
63, 5syl6bi 231 . . 3  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
72, 6sylbir 216 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  ( A. x
( ph  <->  x  =  y
)  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
81, 7mpcom 37 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 187   A.wal 1435    = wceq 1437   E.wex 1659   E!weu 2265   [.wsbc 3299   iotacio 5560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-v 3083  df-sbc 3300  df-un 3441  df-sn 3997  df-pr 3999  df-uni 4217  df-iota 5562
This theorem is referenced by: (None)
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