MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sb6 Unicode version

Theorem 2sb6 2065
Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb6  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
Distinct variable groups:    x, y,
z    y, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 2sb6
StepHypRef Expression
1 sb6 2051 . 2  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x ( x  =  z  ->  [ w  /  y ] ph ) )
2 19.21v 1843 . . . 4  |-  ( A. y ( x  =  z  ->  ( y  =  w  ->  ph )
)  <->  ( x  =  z  ->  A. y
( y  =  w  ->  ph ) ) )
3 impexp 433 . . . . 5  |-  ( ( ( x  =  z  /\  y  =  w )  ->  ph )  <->  ( x  =  z  ->  ( y  =  w  ->  ph )
) )
43albii 1556 . . . 4  |-  ( A. y ( ( x  =  z  /\  y  =  w )  ->  ph )  <->  A. y ( x  =  z  ->  ( y  =  w  ->  ph )
) )
5 sb6 2051 . . . . 5  |-  ( [ w  /  y ]
ph 
<-> 
A. y ( y  =  w  ->  ph )
)
65imbi2i 303 . . . 4  |-  ( ( x  =  z  ->  [ w  /  y ] ph )  <->  ( x  =  z  ->  A. y
( y  =  w  ->  ph ) ) )
72, 4, 63bitr4ri 269 . . 3  |-  ( ( x  =  z  ->  [ w  /  y ] ph )  <->  A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
87albii 1556 . 2  |-  ( A. x ( x  =  z  ->  [ w  /  y ] ph ) 
<-> 
A. x A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
91, 8bitri 240 1  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   [wsb 1638
This theorem is referenced by:  2eu6  2241
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
  Copyright terms: Public domain W3C validator