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Theorem 2ax6elem 2584
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2393 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 39274. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 2393 . . . 4 𝑧 𝑧 = 𝑥
2 nfnae 2458 . . . . . 6 𝑧 ¬ ∀𝑤 𝑤 = 𝑧
3 nfnae 2458 . . . . . 6 𝑧 ¬ ∀𝑤 𝑤 = 𝑥
42, 3nfan 1975 . . . . 5 𝑧(¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥)
5 nfeqf 2444 . . . . . 6 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → Ⅎ𝑤 𝑧 = 𝑥)
6 pm3.21 463 . . . . . 6 (𝑤 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑤 = 𝑦)))
75, 6spimed 2398 . . . . 5 ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → (𝑧 = 𝑥 → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
84, 7eximd 2230 . . . 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 449 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → (¬ ∀𝑤 𝑤 = 𝑥 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)))
11 ax6e 2393 . . 3 𝑧 𝑧 = 𝑦
12 nfae 2456 . . . 4 𝑧𝑤 𝑤 = 𝑥
13 equvini 2481 . . . . 5 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑤 = 𝑦))
14 equtrr 2102 . . . . . . 7 (𝑤 = 𝑥 → (𝑧 = 𝑤𝑧 = 𝑥))
1514anim1d 589 . . . . . 6 (𝑤 = 𝑥 → ((𝑧 = 𝑤𝑤 = 𝑦) → (𝑧 = 𝑥𝑤 = 𝑦)))
1615aleximi 1906 . . . . 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 2230 . . 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 172 1 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857
This theorem is referenced by:  2ax6e  2585
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