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Theorem 2ax6eOLD 2497
Description: Obsolete version of 2ax6e 2496 as of 3-Oct-2023. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2ax6eOLD 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
Distinct variable group:   𝑧,𝑤

Proof of Theorem 2ax6eOLD
StepHypRef Expression
1 aeveq 2062 . . . 4 (∀𝑤 𝑤 = 𝑧𝑧 = 𝑥)
2 aeveq 2062 . . . 4 (∀𝑤 𝑤 = 𝑧𝑤 = 𝑦)
31, 2jca 515 . . 3 (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥𝑤 = 𝑦))
4 19.8a 2182 . . 3 ((𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥𝑤 = 𝑦))
5 19.8a 2182 . . 3 (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
63, 4, 53syl 18 . 2 (∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
7 2ax6elem 2495 . 2 (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))
86, 7pm2.61i 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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
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: (None)
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