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Theorem axc11n11 31694
 Description: Proof of axc11n 2199 from { ax-1 6-- ax-7 1885, axc11 2206 } . Almost identical to axc11nfromc11 33119. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
axc11n11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11n11
StepHypRef Expression
1 axc11 2206 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 49 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 1894 . 2 (𝑥 = 𝑦𝑦 = 𝑥)
42, 3sylg 1725 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137 This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699 This theorem is referenced by: (None)
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