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Theorem 2ax6e 2486
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2485 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.)
Assertion
Ref Expression
2ax6e 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Distinct variable group:   𝑧,𝑤

Proof of Theorem 2ax6e
StepHypRef Expression
1 aeveq 2052 . . . . 5 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 2052 . . . . 5 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 512 . . . 4 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4319.8ad 2171 . . 3 (∀𝑤 𝑤 = 𝑧 → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5419.8ad 2171 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
6 2ax6elem 2485 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
75, 6pm2.61i 183 1 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776
This theorem is referenced by:  2sb5rf  2488  2sb6rf  2489  2sb6rfOLD  2490
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