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Theorem sbcom2v 1909
Description: Lemma for proving sbcom2 1911. It is the same as sbcom2 1911 but with additional distinct variable constraints on  x and  y, and on  w and  z. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2v  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    x, w, z   
x, y, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem sbcom2v
StepHypRef Expression
1 alcom 1412 . . 3  |-  ( A. z A. x ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. x A. z ( x  =  y  ->  ( z  =  w  ->  ph )
) )
2 bi2.04 246 . . . . . 6  |-  ( ( x  =  y  -> 
( z  =  w  ->  ph ) )  <->  ( z  =  w  ->  ( x  =  y  ->  ph )
) )
32albii 1404 . . . . 5  |-  ( A. x ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  A. x ( z  =  w  ->  (
x  =  y  ->  ph ) ) )
4 19.21v 1801 . . . . 5  |-  ( A. x ( z  =  w  ->  ( x  =  y  ->  ph )
)  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
53, 4bitri 182 . . . 4  |-  ( A. x ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
65albii 1404 . . 3  |-  ( A. z A. x ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. z
( z  =  w  ->  A. x ( x  =  y  ->  ph )
) )
7 19.21v 1801 . . . 4  |-  ( A. z ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) )
87albii 1404 . . 3  |-  ( A. x A. z ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. x
( x  =  y  ->  A. z ( z  =  w  ->  ph )
) )
91, 6, 83bitr3i 208 . 2  |-  ( A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) )  <->  A. x
( x  =  y  ->  A. z ( z  =  w  ->  ph )
) )
10 sb6 1814 . . 3  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  A. z ( z  =  w  ->  [ y  /  x ] ph ) )
11 sb6 1814 . . . . 5  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
1211imbi2i 224 . . . 4  |-  ( ( z  =  w  ->  [ y  /  x ] ph )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
1312albii 1404 . . 3  |-  ( A. z ( z  =  w  ->  [ y  /  x ] ph )  <->  A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
1410, 13bitri 182 . 2  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
15 sb6 1814 . . 3  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  A. x ( x  =  y  ->  [ w  /  z ] ph ) )
16 sb6 1814 . . . . 5  |-  ( [ w  /  z ]
ph 
<-> 
A. z ( z  =  w  ->  ph )
)
1716imbi2i 224 . . . 4  |-  ( ( x  =  y  ->  [ w  /  z ] ph )  <->  ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) )
1817albii 1404 . . 3  |-  ( A. x ( x  =  y  ->  [ w  /  z ] ph ) 
<-> 
A. x ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) )
1915, 18bitri 182 . 2  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  A. x ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) )
209, 14, 193bitr4i 210 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  sbcom2v2  1910
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