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Theorem pm13.192 26777
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 191 . . . . . . 7  |-  ( ( x  =  A  <->  x  =  y )  ->  (
x  =  y  ->  x  =  A )
)
21alimi 1546 . . . . . 6  |-  ( A. x ( x  =  A  <->  x  =  y
)  ->  A. x
( x  =  y  ->  x  =  A ) )
3 nfv 1629 . . . . . . 7  |-  F/ x  y  =  A
4 eqeq1 2259 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
53, 4equsal 1850 . . . . . 6  |-  ( A. x ( x  =  y  ->  x  =  A )  <->  y  =  A )
62, 5sylib 190 . . . . 5  |-  ( A. x ( x  =  A  <->  x  =  y
)  ->  y  =  A )
7 eqeq2 2262 . . . . . . 7  |-  ( A  =  y  ->  (
x  =  A  <->  x  =  y ) )
87eqcoms 2256 . . . . . 6  |-  ( y  =  A  ->  (
x  =  A  <->  x  =  y ) )
98alrimiv 2012 . . . . 5  |-  ( y  =  A  ->  A. x
( x  =  A  <-> 
x  =  y ) )
106, 9impbii 182 . . . 4  |-  ( A. x ( x  =  A  <->  x  =  y
)  <->  y  =  A )
1110anbi1i 679 . . 3  |-  ( ( A. x ( x  =  A  <->  x  =  y )  /\  ph ) 
<->  ( y  =  A  /\  ph ) )
1211exbii 1580 . 2  |-  ( E. y ( A. x
( x  =  A  <-> 
x  =  y )  /\  ph )  <->  E. y
( y  =  A  /\  ph ) )
13 sbc5 2945 . 2  |-  ( [. A  /  y ]. ph  <->  E. y
( y  =  A  /\  ph ) )
1412, 13bitr4i 245 1  |-  ( E. y ( A. x
( x  =  A  <-> 
x  =  y )  /\  ph )  <->  [. A  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619   [.wsbc 2921
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-sbc 2922
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