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Theorem ax6e2eq 36367
Description: Alternate form of ax6e 2031 for non-distinct  x,  y and  u  =  v. ax6e2eq 36367 is derived from ax6e2eqVD 36751. (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 1775 . . . . . . 7  |-  E. x  x  =  u
2 hbae 2083 . . . . . . . 8  |-  ( A. x  x  =  y  ->  A. x A. x  x  =  y )
3 ax-7 1816 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  u  -> 
y  =  u ) )
43sps 1891 . . . . . . . . 9  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  y  =  u ) )
54ancld 553 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( x  =  u  ->  ( x  =  u  /\  y  =  u ) ) )
62, 5eximdh 1696 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( E. x  x  =  u  ->  E. x
( x  =  u  /\  y  =  u ) ) )
71, 6mpi 21 . . . . . 6  |-  ( A. x  x  =  y  ->  E. x ( x  =  u  /\  y  =  u ) )
87axc4i 1928 . . . . 5  |-  ( A. x  x  =  y  ->  A. x E. x
( x  =  u  /\  y  =  u ) )
9 axc11 2082 . . . . 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 1777 . . . 4  |-  ( A. y E. x ( x  =  u  /\  y  =  u )  ->  E. y E. x ( x  =  u  /\  y  =  u ) )
1210, 11syl 17 . . 3  |-  ( A. x  x  =  y  ->  E. y E. x
( x  =  u  /\  y  =  u ) )
13 excomim 1876 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  u )  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
1412, 13syl 17 . 2  |-  ( A. x  x  =  y  ->  E. x E. y
( x  =  u  /\  y  =  u ) )
15 equtrr 1823 . . . 4  |-  ( u  =  v  ->  (
y  =  u  -> 
y  =  v ) )
1615anim2d 565 . . 3  |-  ( u  =  v  ->  (
( x  =  u  /\  y  =  u )  ->  ( x  =  u  /\  y  =  v ) ) )
17162eximdv 1735 . 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 1405   E.wex 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028
This theorem depends on definitions:  df-bi 187  df-an 371  df-ex 1636  df-nf 1640
This theorem is referenced by:  ax6e2ndeq  36369  ax6e2ndeqVD  36753  ax6e2ndeqALT  36775
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