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Theorem iotavalsb 27736
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 1730 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) )
2 df-eu 2160 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 iotavalb 27733 . . . 4  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
4 dfsbcq 3006 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
54eqcoms 2299 . . . 4  |-  ( ( iota x ph )  =  y  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) )
63, 5syl6bi 219 . . 3  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
72, 6sylbir 204 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  ( A. x
( ph  <->  x  =  y
)  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps ) ) )
81, 7mpcom 32 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 176   A.wal 1530   E.wex 1531    = wceq 1632   E!weu 2156   [.wsbc 3004   iotacio 5233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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