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Theorem pm13.192 27520
Description: Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
Assertion
Ref Expression
pm13.192  |-  ( E. y ( A. x
( x  =  A  <-> 
x  =  y )  /\  ph )  <->  [. A  / 
y ]. ph )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem pm13.192
StepHypRef Expression
1 bi2 190 . . . . . . 7  |-  ( ( x  =  A  <->  x  =  y )  ->  (
x  =  y  ->  x  =  A )
)
21alimi 1568 . . . . . 6  |-  ( A. x ( x  =  A  <->  x  =  y
)  ->  A. x
( x  =  y  ->  x  =  A ) )
3 nfv 1629 . . . . . . 7  |-  F/ x  y  =  A
4 eqeq1 2436 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
53, 4equsal 1999 . . . . . 6  |-  ( A. x ( x  =  y  ->  x  =  A )  <->  y  =  A )
62, 5sylib 189 . . . . 5  |-  ( A. x ( x  =  A  <->  x  =  y
)  ->  y  =  A )
7 eqeq2 2439 . . . . . . 7  |-  ( A  =  y  ->  (
x  =  A  <->  x  =  y ) )
87eqcoms 2433 . . . . . 6  |-  ( y  =  A  ->  (
x  =  A  <->  x  =  y ) )
98alrimiv 1641 . . . . 5  |-  ( y  =  A  ->  A. x
( x  =  A  <-> 
x  =  y ) )
106, 9impbii 181 . . . 4  |-  ( A. x ( x  =  A  <->  x  =  y
)  <->  y  =  A )
1110anbi1i 677 . . 3  |-  ( ( A. x ( x  =  A  <->  x  =  y )  /\  ph ) 
<->  ( y  =  A  /\  ph ) )
1211exbii 1592 . 2  |-  ( E. y ( A. x
( x  =  A  <-> 
x  =  y )  /\  ph )  <->  E. y
( y  =  A  /\  ph ) )
13 sbc5 3172 . 2  |-  ( [. A  /  y ]. ph  <->  E. y
( y  =  A  /\  ph ) )
1412, 13bitr4i 244 1  |-  ( E. y ( A. x
( x  =  A  <-> 
x  =  y )  /\  ph )  <->  [. A  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   [.wsbc 3148
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 2411
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-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-sbc 3149
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