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Theorem 2ax6e 2495
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2494 with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) (New usage is discouraged.)
Assertion
Ref Expression
2ax6e 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Distinct variable group:   𝑧,𝑤

Proof of Theorem 2ax6e
StepHypRef Expression
1 aeveq 2061 . . . . 5 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 2061 . . . . 5 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 515 . . . 4 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4319.8ad 2182 . . 3 (∀𝑤 𝑤 = 𝑧 → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5419.8ad 2182 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
6 2ax6elem 2494 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
75, 6pm2.61i 185 1 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  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 2145  ax-11 2161  ax-12 2178  ax-13 2391
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:  2sb5rf  2497  2sb6rf  2498
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