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Theorem 2ax6e 2438
 Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2437 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 1969 . . . 4 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 1969 . . . 4 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 553 . . 3 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4 19.8a 2039 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5 19.8a 2039 . . 3 (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
63, 4, 53syl 18 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
7 2ax6elem 2437 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
86, 7pm2.61i 175 1 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  2sb5rf  2439  2sb6rf  2440
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