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Theorem 2pm13.193 36962
Description: pm13.193 36805 for two variables. pm13.193 36805 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 37339. (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 765 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  x  =  u )
2 simplr 767 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  y  =  v )
3 simpr 467 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ u  /  x ] [ v  / 
y ] ph )
4 sbequ2 1809 . . . . 5  |-  ( x  =  u  ->  ( [ u  /  x ] [ v  /  y ] ph  ->  [ v  /  y ] ph ) )
51, 3, 4sylc 62 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  [ v  / 
y ] ph )
6 sbequ2 1809 . . . 4  |-  ( y  =  v  ->  ( [ v  /  y ] ph  ->  ph ) )
72, 5, 6sylc 62 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ph )
81, 2, 7jca31 541 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
9 simpll 765 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  x  =  u )
10 simplr 767 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  y  =  v )
11 simpr 467 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ph )
12 sbequ1 2092 . . . . 5  |-  ( y  =  v  ->  ( ph  ->  [ v  / 
y ] ph )
)
1310, 11, 12sylc 62 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ v  /  y ] ph )
14 sbequ1 2092 . . . 4  |-  ( x  =  u  ->  ( [ v  /  y ] ph  ->  [ u  /  x ] [ v  /  y ] ph ) )
159, 13, 14sylc 62 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  [ u  /  x ] [ v  /  y ] ph )
169, 10, 15jca31 541 . 2  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( ( x  =  u  /\  y  =  v )  /\  [
u  /  x ] [ v  /  y ] ph ) )
178, 16impbii 192 1  |-  ( ( ( x  =  u  /\  y  =  v )  /\  [ u  /  x ] [ v  /  y ] ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-sb 1808
This theorem is referenced by:  2sb5nd  36970  2sb5ndVD  37346  2sb5ndALT  37368
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