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Theorem 2ax6elem 2482
 Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2390 instances ∃𝑧𝑧 = 𝑥 and ∃𝑤𝑤 = 𝑦 into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 41637. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Wolf Lammen, 27-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
2ax6elem (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 2390 . . . 4 𝑧 𝑧 = 𝑥
2 nfnae 2445 . . . . . 6 𝑧 ¬ ∀𝑤 𝑤 = 𝑧
3 nfnae 2445 . . . . . 6 𝑧 ¬ ∀𝑤 𝑤 = 𝑥
42, 3nfan 1900 . . . . 5 𝑧(¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥)
5 nfeqf 2388 . . . . . 6 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → Ⅎ𝑤 𝑧 = 𝑥)
6 pm3.21 475 . . . . . 6 (𝑤 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑤 = 𝑦)))
75, 6spimed 2395 . . . . 5 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → (𝑧 = 𝑥 → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
84, 7eximd 2214 . . . 4 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → (∃𝑧 𝑧 = 𝑥 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
91, 8mpi 20 . . 3 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
109ex 416 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → (¬ ∀𝑤 𝑤 = 𝑥 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
11 ax6e 2390 . . 3 𝑧 𝑧 = 𝑦
12 nfae 2444 . . . 4 𝑧𝑤 𝑤 = 𝑥
13 equvini 2466 . . . . 5 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑤 = 𝑦))
14 equtrr 2029 . . . . . . 7 (𝑤 = 𝑥 → (𝑧 = 𝑤𝑧 = 𝑥))
1514anim1d 613 . . . . . 6 (𝑤 = 𝑥 → ((𝑧 = 𝑤𝑤 = 𝑦) → (𝑧 = 𝑥𝑤 = 𝑦)))
1615aleximi 1833 . . . . 5 (∀𝑤 𝑤 = 𝑥 → (∃𝑤(𝑧 = 𝑤𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
1713, 16syl5 34 . . . 4 (∀𝑤 𝑤 = 𝑥 → (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
1812, 17eximd 2214 . . 3 (∀𝑤 𝑤 = 𝑥 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
1911, 18mpi 20 . 2 (∀𝑤 𝑤 = 𝑥 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
2010, 19pm2.61d2 184 1 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  2ax6e  2483  2ax6eOLD  2484
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