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Theorem pm14.12 36674
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 2296 . 2  |-  ( E! x ph  ->  E* x ph )
2 nfv 1752 . . . 4  |-  F/ y
ph
32mo3 2304 . . 3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
4 sbsbc 3304 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
54anbi2i 699 . . . . 5  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  [. y  /  x ]. ph ) )
65imbi1i 327 . . . 4  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y )
)
762albii 1689 . . 3  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
83, 7bitri 253 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
91, 8sylib 200 1  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1436   [wsb 1787   E!weu 2266   E*wmo 2267   [.wsbc 3300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-sbc 3301
This theorem is referenced by:  pm14.24  36685
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