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Theorem 2pm13.193 27334
Description: pm13.193 26944 for two variables. pm13.193 26944 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 27692. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2pm13.193  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )

Proof of Theorem 2pm13.193
StepHypRef Expression
1 simpll 733 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  x  =  u )
2 simplr 734 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  y  =  v )
3 simpr 449 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
4 sbequ2 1891 . . . . 5  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
51, 3, 4sylc 58 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ v  / 
y ] ph )
6 sbequ2 1891 . . . 4  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
72, 5, 6sylc 58 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ph )
81, 2, 7jca31 522 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
9 simpll 733 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  x  =  u )
10 simplr 734 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  y  =  v )
11 simpr 449 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
12 sbequ1 1890 . . . . 5  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
1310, 11, 12sylc 58 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ v  /  y ] ph )
14 sbequ1 1890 . . . 4  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
159, 13, 14sylc 58 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ u  /  x ] [ v  /  y ] ph )
169, 10, 15jca31 522 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
178, 16impbii 182 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   [wsb 1883
This theorem is referenced by:  2sb5nd  27342  2sb5ndVD  27699  2sb5ndALT  27722
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-sb 1884
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