Theorem axc11n11 37307

Description: Proof of axc11n 2458 from { ax-1 6-- ax-7 2038, axc11 2462 } . Almost identical to axc11nfromc11 39700. (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 2462	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 53	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2047	. 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	2, 3	sylg 1853	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)