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Theorem 2ax6e 2598
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2597 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.)
Assertion
Ref Expression
2ax6e 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Distinct variable group:   𝑧,𝑤

Proof of Theorem 2ax6e
StepHypRef Expression
1 aeveq 2139 . . . 4 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 2139 . . . 4 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 501 . . 3 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4 19.8a 2206 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5 19.8a 2206 . . 3 (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
63, 4, 53syl 18 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
7 2ax6elem 2597 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
86, 7pm2.61i 176 1 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858
This theorem is referenced by:  2sb5rf  2599  2sb6rf  2600
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