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Theorem axc11n11 36068
Description: Proof of axc11n 2419 from { ax-1 6-- ax-7 2003, axc11 2423 } . Almost identical to axc11nfromc11 38309. (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 2423 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 2012 . 2 (𝑥 = 𝑦𝑦 = 𝑥)
42, 3sylg 1817 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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