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Theorem 2pm13.193 28318
Description: pm13.193 27611 for two variables. pm13.193 27611 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 28679. (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 730 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  x  =  u )
2 simplr 731 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  y  =  v )
3 simpr 447 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
4 sbequ2 1631 . . . . 5  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
51, 3, 4sylc 56 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ v  / 
y ] ph )
6 sbequ2 1631 . . . 4  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
72, 5, 6sylc 56 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ph )
81, 2, 7jca31 520 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
9 simpll 730 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  x  =  u )
10 simplr 731 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  y  =  v )
11 simpr 447 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
12 sbequ1 1859 . . . . 5  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
1310, 11, 12sylc 56 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ v  /  y ] ph )
14 sbequ1 1859 . . . 4  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
159, 13, 14sylc 56 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ u  /  x ] [ v  /  y ] ph )
169, 10, 15jca31 520 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
178, 16impbii 180 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   [wsb 1629
This theorem is referenced by:  2sb5nd  28326  2sb5ndVD  28686  2sb5ndALT  28709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630
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