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Theorem 2sb6 2159
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 2142 . 2  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x ( x  =  z  ->  [ w  /  y ] ph ) )
2 19.21v 1921 . . . 4  |-  ( A. y ( x  =  z  ->  ( y  =  w  ->  ph )
)  <->  ( x  =  z  ->  A. y
( y  =  w  ->  ph ) ) )
3 impexp 446 . . . . 5  |-  ( ( ( x  =  z  /\  y  =  w )  ->  ph )  <->  ( x  =  z  ->  ( y  =  w  ->  ph )
) )
43albii 1611 . . . 4  |-  ( A. y ( ( x  =  z  /\  y  =  w )  ->  ph )  <->  A. y ( x  =  z  ->  ( y  =  w  ->  ph )
) )
5 sb6 2142 . . . . 5  |-  ( [ w  /  y ]
ph 
<-> 
A. y ( y  =  w  ->  ph )
)
65imbi2i 312 . . . 4  |-  ( ( x  =  z  ->  [ w  /  y ] ph )  <->  ( x  =  z  ->  A. y
( y  =  w  ->  ph ) ) )
72, 4, 63bitr4ri 278 . . 3  |-  ( ( x  =  z  ->  [ w  /  y ] ph )  <->  A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
87albii 1611 . 2  |-  ( A. x ( x  =  z  ->  [ w  /  y ] ph ) 
<-> 
A. x A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
91, 8bitri 249 1  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
( ( x  =  z  /\  y  =  w )  ->  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  sbcom2  2160  2exsb  2191  2moOLD  2370  2eu6  2380  2eu6OLD  2381
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