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Theorem mo4f 2314
Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
Hypotheses
Ref Expression
mo4f.1  |-  F/ x ps
mo4f.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
mo4f  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem mo4f
StepHypRef Expression
1 nfv 1630 . . 3  |-  F/ y
ph
21mo3 2313 . 2  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
3 mo4f.1 . . . . . 6  |-  F/ x ps
4 mo4f.2 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
53, 4sbie 2150 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65anbi2i 677 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
76imbi1i 317 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
872albii 1577 . 2  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
92, 8bitri 242 1  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   F/wnf 1554   [wsb 1659   E*wmo 2283
This theorem is referenced by:  mo4  2315  mob2  3115  moop2  4452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287
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