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Theorem a9e2eq 28323
Description: Alternate form of a9e 1891 for non-distinct  x,  y and  u  =  v. a9e2eq 28323 is derived from a9e2eqVD 28683. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9e2eq  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v

Proof of Theorem a9e2eq
StepHypRef Expression
1 a9ev 1637 . . . . . . 7  |-  E. x  x  =  u
2 hbae 1893 . . . . . . . 8  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
3 ax-8 1643 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
43sps 1739 . . . . . . . . 9  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  y  =  u ) )
54ancld 536 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
62, 5eximdh 1575 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) )
71, 6mpi 16 . . . . . 6  |-  ( A. x  x  =  y  ->  E. x ( x  =  u  /\  y  =  u ) )
87a5i 1758 . . . . 5  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
9 ax10o 1892 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. x E. x ( x  =  u  /\  y  =  u )  ->  A. y E. x ( x  =  u  /\  y  =  u ) ) )
108, 9mpd 14 . . . 4  |-  ( A. x  x  =  y  ->  A. y E. x
( x  =  u  /\  y  =  u ) )
11 19.2 1671 . . . 4  |-  ( A. y E. x ( x  =  u  /\  y  =  u )  ->  E. y E. x ( x  =  u  /\  y  =  u ) )
1210, 11syl 15 . . 3  |-  ( A. x  x  =  y  ->  E. y E. x
( x  =  u  /\  y  =  u ) )
13 excomim 1785 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
1412, 13syl 15 . 2  |-  ( A. x  x  =  y  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
15 equtrr 1653 . . . 4  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
1615anim2d 548 . . 3  |-  ( u  =  v  ->  (
( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) )
17162eximdv 1610 . 2  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
1814, 17syl5com 26 1  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  a9e2ndeq  28325  a9e2ndeqVD  28685  a9e2ndeqALT  28708
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532
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