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Theorem bj-ax6e 32778
Description: Proof of ax6e 2286 (hence ax6 2287) from Tarski's system, ax-c9 34494, ax-c16 34496. Remark: ax-6 1945 is used only via its principal (unbundled) instance ax6v 1946. (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 1949 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
21a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3 bj-ax6elem1 32776 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4 bj-ax6elem2 32777 . . . 4 (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
53, 4syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
62, 5pm2.61i 176 . 2 (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7 ax6evr 1988 . 2 𝑧 𝑦 = 𝑧
86, 7exlimiiv 1899 1 𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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