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Theorem ax6e2eq 33473
Description: Alternate form of ax6e 2003 for non-distinct  x,  y and  u  =  v. ax6e2eq 33473 is derived from ax6e2eqVD 33850. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2eq  |-  ( 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 ax6e2eq
StepHypRef Expression
1 ax6ev 1750 . . . . . . 7  |-  E. x  x  =  u
2 hbae 2056 . . . . . . . 8  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
3 ax-7 1791 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
43sps 1866 . . . . . . . . 9  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  y  =  u ) )
54ancld 553 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
62, 5eximdh 1674 . . . . . . 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 ) )
87axc4i 1899 . . . . 5  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
9 axc11 2055 . . . . 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 1752 . . . 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 1851 . . 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 1798 . . . 4  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
1615anim2d 565 . . 3  |-  ( u  =  v  ->  (
( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) )
17162eximdv 1713 . 2  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
1814, 17syl5com 30 1  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393   E.wex 1613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618
This theorem is referenced by:  ax6e2ndeq  33475  ax6e2ndeqVD  33852  ax6e2ndeqALT  33874
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