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Theorem 2ax6elem 2193
 Description: We can always find values matching and , as long as they are represented by distinct variables. This theorem merges two ax6e 2002 instances and into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 33331. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 2002 . . . 4
2 nfnae 2058 . . . . . 6
3 nfnae 2058 . . . . . 6
42, 3nfan 1928 . . . . 5
5 nfeqf 2045 . . . . . 6
6 pm3.21 448 . . . . . 6
75, 6spimed 2007 . . . . 5
84, 7eximd 1882 . . . 4
91, 8mpi 17 . . 3
109ex 434 . 2
11 ax6e 2002 . . 3
12 nfae 2056 . . . 4
13 equvini 2087 . . . . 5
14 equtrr 1797 . . . . . . 7
1514anim1d 564 . . . . . 6
1615aleximi 1653 . . . . 5
1713, 16syl5 32 . . . 4
1812, 17eximd 1882 . . 3
1911, 18mpi 17 . 2
2010, 19pm2.61d2 160 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612 This theorem is referenced by:  2ax6e  2194 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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