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Theorem bj-ax6e 34075
 Description: Proof of ax6e 2402 (hence ax6 2403) from Tarski's system, ax-c9 36144, ax-c16 36146. Remark: ax-6 1970 is used only via its principal (unbundled) instance ax6v 1971. (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 1981 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
21a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
3 bj-ax6elem1 34073 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
4 bj-ax6elem2 34074 . . . 4 (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
53, 4syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦))
62, 5pm2.61i 185 . 2 (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
7 ax6evr 2022 . 2 𝑧 𝑦 = 𝑧
86, 7exlimiiv 1932 1 𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀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-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: (None)
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