Theorem axc11n11 36716

Description: Proof of axc11n 2426 from { ax-1 6-- ax-7 2009, axc11 2430 } . Almost identical to axc11nfromc11 38965. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc11n11	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11n11

Step	Hyp	Ref	Expression
1		axc11 2430	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2018	. 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	2, 3	sylg 1824	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)