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Theorem 2ax6elem 2244
Description: We can always find values matching  x and  y, as long as they are represented by distinct variables. This theorem merges two ax6e 2056 instances  E. z z  =  x and  E. w w  =  y into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 36783. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem  |-  ( -. 
A. w  w  =  z  ->  E. z E. w ( z  =  x  /\  w  =  y ) )

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 2056 . . . 4  |-  E. z 
z  =  x
2 nfnae 2113 . . . . . 6  |-  F/ z  -.  A. w  w  =  z
3 nfnae 2113 . . . . . 6  |-  F/ z  -.  A. w  w  =  x
42, 3nfan 1984 . . . . 5  |-  F/ z ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )
5 nfeqf 2100 . . . . . 6  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  F/ w  z  =  x )
6 pm3.21 449 . . . . . 6  |-  ( w  =  y  ->  (
z  =  x  -> 
( z  =  x  /\  w  =  y ) ) )
75, 6spimed 2061 . . . . 5  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  ( z  =  x  ->  E. w
( z  =  x  /\  w  =  y ) ) )
84, 7eximd 1933 . . . 4  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  ( E. z  z  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) ) )
91, 8mpi 21 . . 3  |-  ( ( -.  A. w  w  =  z  /\  -.  A. w  w  =  x )  ->  E. z E. w ( z  =  x  /\  w  =  y ) )
109ex 435 . 2  |-  ( -. 
A. w  w  =  z  ->  ( -.  A. w  w  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) ) )
11 ax6e 2056 . . 3  |-  E. z 
z  =  y
12 nfae 2111 . . . 4  |-  F/ z A. w  w  =  x
13 equvini 2142 . . . . 5  |-  ( z  =  y  ->  E. w
( z  =  w  /\  w  =  y ) )
14 equtrr 1847 . . . . . . 7  |-  ( w  =  x  ->  (
z  =  w  -> 
z  =  x ) )
1514anim1d 566 . . . . . 6  |-  ( w  =  x  ->  (
( z  =  w  /\  w  =  y )  ->  ( z  =  x  /\  w  =  y ) ) )
1615aleximi 1700 . . . . 5  |-  ( A. w  w  =  x  ->  ( E. w ( z  =  w  /\  w  =  y )  ->  E. w ( z  =  x  /\  w  =  y ) ) )
1713, 16syl5 33 . . . 4  |-  ( A. w  w  =  x  ->  ( z  =  y  ->  E. w ( z  =  x  /\  w  =  y ) ) )
1812, 17eximd 1933 . . 3  |-  ( A. w  w  =  x  ->  ( E. z  z  =  y  ->  E. z E. w ( z  =  x  /\  w  =  y ) ) )
1911, 18mpi 21 . 2  |-  ( A. w  w  =  x  ->  E. z E. w
( z  =  x  /\  w  =  y ) )
2010, 19pm2.61d2 163 1  |-  ( -. 
A. w  w  =  z  ->  E. z E. w ( z  =  x  /\  w  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  2ax6e  2245
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