Theorem axc11n11 36192

Description: Proof of axc11n 2420 from { ax-1 6-- ax-7 2003, axc11 2424 } . Almost identical to axc11nfromc11 38430. (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 2424	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2012	. 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	2, 3	sylg 1817	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

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 2166  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)