Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ax6elem Structured version   Visualization version   Unicode version

Theorem 2ax6elem 2278
 Description: We can always find values matching and , as long as they are represented by distinct variables. This theorem merges two ax6e 2094 instances and into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 36925. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 2094 . . . 4
2 nfnae 2152 . . . . . 6
3 nfnae 2152 . . . . . 6
42, 3nfan 2011 . . . . 5
5 nfeqf 2139 . . . . . 6
6 pm3.21 450 . . . . . 6
75, 6spimed 2099 . . . . 5
84, 7eximd 1960 . . . 4
91, 8mpi 20 . . 3
109ex 436 . 2
11 ax6e 2094 . . 3
12 nfae 2150 . . . 4
13 equvini 2179 . . . . 5
14 equtrr 1866 . . . . . . 7
1514anim1d 568 . . . . . 6
1615aleximi 1704 . . . . 5
1713, 16syl5 33 . . . 4
1812, 17eximd 1960 . . 3
1911, 18mpi 20 . 2
2010, 19pm2.61d2 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1442  wex 1663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091 This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668 This theorem is referenced by:  2ax6e  2279
 Copyright terms: Public domain W3C validator