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Theorem 2ax6e 2449
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2448 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 1981 . . . 4 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 1981 . . . 4 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 554 . . 3 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4 19.8a 2051 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5 19.8a 2051 . . 3 (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
63, 4, 53syl 18 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
7 2ax6elem 2448 . 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 384  wal 1480  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709
This theorem is referenced by:  2sb5rf  2450  2sb6rf  2451
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