MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2mos Unicode version

Theorem 2mos 2333
Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
Hypothesis
Ref Expression
2mos.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
2mos  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Distinct variable groups:    z, w, ph    x, y, ps    x, z, w, y
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem 2mos
StepHypRef Expression
1 2mo 2332 . 2  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
) )
2 nfv 1626 . . . . . . 7  |-  F/ x ps
3 2mos.1 . . . . . . . 8  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
43sbiedv 2086 . . . . . . 7  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  ps ) )
52, 4sbie 2087 . . . . . 6  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  ps )
65anbi2i 676 . . . . 5  |-  ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ph  /\  ps ) )
76imbi1i 316 . . . 4  |-  ( ( ( ph  /\  [
z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
872albii 1573 . . 3  |-  ( A. z A. w ( (
ph  /\  [ z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w ) )  <->  A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
982albii 1573 . 2  |-  ( A. x A. y A. z A. w ( ( ph  /\ 
[ z  /  x ] [ w  /  y ] ph )  ->  (
x  =  z  /\  y  =  w )
)  <->  A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
101, 9bitri 241 1  |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\ 
ps )  ->  (
x  =  z  /\  y  =  w )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
  Copyright terms: Public domain W3C validator