Theorem axc11n11 37364

Description: Proof of axc11n 2460 from { ax-1 6-- ax-7 2041, axc11 2464 } . Almost identical to axc11nfromc11 39758. (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 2464	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 53	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 2050	. 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	2, 3	sylg 1856	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by: (None)