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Theorem pm14.12 27724
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 2196 . 2  |-  ( E! x ph  ->  E* x ph )
2 nfv 1609 . . . 4  |-  F/ y
ph
32mo3 2187 . . 3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
4 sbsbc 3008 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
54anbi2i 675 . . . . 5  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  [. y  /  x ]. ph ) )
65imbi1i 315 . . . 4  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y )
)
762albii 1557 . . 3  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
83, 7bitri 240 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
91, 8sylib 188 1  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   [wsb 1638   E!weu 2156   E*wmo 2157   [.wsbc 3004
This theorem is referenced by:  pm14.24  27735
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-sbc 3005
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