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Theorem a9e2eq 27980
Description: Alternate form of a9e 1941 for non-distinct  x,  y and  u  =  v. a9e2eq 27980 is derived from a9e2eqVD 28353. (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 1663 . . . . . . 7  |-  E. x  x  =  u
2 hbae 1984 . . . . . . . 8  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
3 ax-8 1682 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
43sps 1762 . . . . . . . . 9  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  y  =  u ) )
54ancld 537 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
62, 5eximdh 1595 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) )
71, 6mpi 17 . . . . . 6  |-  ( A. x  x  =  y  ->  E. x ( x  =  u  /\  y  =  u ) )
87a5i 1797 . . . . 5  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
9 ax10o 1983 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. x E. x ( x  =  u  /\  y  =  u )  ->  A. y E. x ( x  =  u  /\  y  =  u ) ) )
108, 9mpd 15 . . . 4  |-  ( A. x  x  =  y  ->  A. y E. x
( x  =  u  /\  y  =  u ) )
11 19.2 1666 . . . 4  |-  ( A. y E. x ( x  =  u  /\  y  =  u )  ->  E. y E. x ( x  =  u  /\  y  =  u ) )
1210, 11syl 16 . . 3  |-  ( A. x  x  =  y  ->  E. y E. x
( x  =  u  /\  y  =  u ) )
13 excomim 1749 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
1412, 13syl 16 . 2  |-  ( A. x  x  =  y  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
15 equtrr 1690 . . . 4  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
1615anim2d 549 . . 3  |-  ( u  =  v  ->  (
( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) )
17162eximdv 1631 . 2  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
1814, 17syl5com 28 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 359   A.wal 1546   E.wex 1547
This theorem is referenced by:  a9e2ndeq  27982  a9e2ndeqVD  28355  a9e2ndeqALT  28378
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator