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Theorem bj-ax6e 31842
 Description: Proof of ax6e 2238 (hence ax6 2239) from Tarski's system, ax-c9 33193, ax-c16 33195. Remark: ax-6 1875 is used only via its principal (unbundled) instance ax6v 1876. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ax6e 𝑥 𝑥 = 𝑦

Proof of Theorem bj-ax6e
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 19.2 1879 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
21a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3 bj-ax6elem1 31840 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4 bj-ax6elem2 31841 . . . 4 (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
53, 4syl6 34 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
62, 5pm2.61i 175 . 2 (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7 ax6evr 1929 . 2 𝑧 𝑦 = 𝑧
86, 7exlimiiv 1846 1 𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀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-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: (None)
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