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Theorem pm14.12 26989
Description: Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
pm14.12  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
Distinct variable groups:    ph, y    x, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm14.12
StepHypRef Expression
1 eumo 2158 . 2  |-  ( E! x ph  ->  E* x ph )
2 nfv 1629 . . . 4  |-  F/ y
ph
32mo3 2149 . . 3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
4 sbsbc 2970 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
54anbi2i 678 . . . . 5  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  [. y  /  x ]. ph ) )
65imbi1i 317 . . . 4  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y )
)
762albii 1555 . . 3  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
83, 7bitri 242 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
91, 8sylib 190 1  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   [wsb 1883   E!weu 2118   E*wmo 2119   [.wsbc 2966
This theorem is referenced by:  pm14.24  27000
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 1927  ax-ext 2239
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-sbc 2967
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