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Theorem 2uasbanh 28363
Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 28363 is derived from 2uasbanhVD 28736. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
2uasbanh.1  |-  ( ch  <->  ( E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
Assertion
Ref Expression
2uasbanh  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  <->  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v
Allowed substitution hints:    ph( x, y, v, u)    ps( x, y, v, u)    ch( x, y, v, u)

Proof of Theorem 2uasbanh
StepHypRef Expression
1 simpl 444 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  -> 
( x  =  u  /\  y  =  v ) )
2 simprl 733 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  ->  ph )
31, 2jca 519 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  -> 
( ( x  =  u  /\  y  =  v )  /\  ph ) )
432eximi 1583 . . 3  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  ->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )
)
5 simprr 734 . . . . 5  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  ->  ps )
61, 5jca 519 . . . 4  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  -> 
( ( x  =  u  /\  y  =  v )  /\  ps ) )
762eximi 1583 . . 3  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  ->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps )
)
84, 7jca 519 . 2  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  -> 
( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
9 2uasbanh.1 . . 3  |-  ( ch  <->  ( E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
109simplbi 447 . . . . . 6  |-  ( ch 
->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph ) )
11 simpl 444 . . . . . . . . . 10  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  ( x  =  u  /\  y  =  v ) )
12112eximi 1583 . . . . . . . . 9  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  ->  E. x E. y
( x  =  u  /\  y  =  v ) )
1310, 12syl 16 . . . . . . . 8  |-  ( ch 
->  E. x E. y
( x  =  u  /\  y  =  v ) )
14 a9e2ndeq 28361 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
1513, 14sylibr 204 . . . . . . 7  |-  ( ch 
->  ( -.  A. x  x  =  y  \/  u  =  v )
)
16 2sb5nd 28362 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  -> 
( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
1715, 16syl 16 . . . . . 6  |-  ( ch 
->  ( [ u  /  x ] [ v  / 
y ] ph  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) ) )
1810, 17mpbird 224 . . . . 5  |-  ( ch 
->  [ u  /  x ] [ v  /  y ] ph )
199simprbi 451 . . . . . 6  |-  ( ch 
->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ps ) )
20 2sb5nd 28362 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  -> 
( [ u  /  x ] [ v  / 
y ] ps  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
2115, 20syl 16 . . . . . 6  |-  ( ch 
->  ( [ u  /  x ] [ v  / 
y ] ps  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
2219, 21mpbird 224 . . . . 5  |-  ( ch 
->  [ u  /  x ] [ v  /  y ] ps )
23 sban 2122 . . . . . . 7  |-  ( [ v  /  y ] ( ph  /\  ps ) 
<->  ( [ v  / 
y ] ph  /\  [ v  /  y ] ps ) )
2423sbbii 1661 . . . . . 6  |-  ( [ u  /  x ] [ v  /  y ] ( ph  /\  ps )  <->  [ u  /  x ] ( [ v  /  y ] ph  /\ 
[ v  /  y ] ps ) )
25 sban 2122 . . . . . 6  |-  ( [ u  /  x ]
( [ v  / 
y ] ph  /\  [ v  /  y ] ps )  <->  ( [
u  /  x ] [ v  /  y ] ph  /\  [ u  /  x ] [ v  /  y ] ps ) )
2624, 25bitri 241 . . . . 5  |-  ( [ u  /  x ] [ v  /  y ] ( ph  /\  ps )  <->  ( [ u  /  x ] [ v  /  y ] ph  /\ 
[ u  /  x ] [ v  /  y ] ps ) )
2718, 22, 26sylanbrc 646 . . . 4  |-  ( ch 
->  [ u  /  x ] [ v  /  y ] ( ph  /\  ps ) )
28 2sb5nd 28362 . . . . 5  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  -> 
( [ u  /  x ] [ v  / 
y ] ( ph  /\ 
ps )  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\  ps )
) ) )
2915, 28syl 16 . . . 4  |-  ( ch 
->  ( [ u  /  x ] [ v  / 
y ] ( ph  /\ 
ps )  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\  ps )
) ) )
3027, 29mpbid 202 . . 3  |-  ( ch 
->  E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ( ph  /\  ps ) ) )
319, 30sylbir 205 . 2  |-  ( ( E. x E. y
( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) )  ->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\  ps )
) )
328, 31impbii 181 1  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ( ph  /\ 
ps ) )  <->  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  /\  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1546   E.wex 1547   [wsb 1655
This theorem is referenced by:  2uasban  28364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-ne 2573  df-v 2922
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