Theorem wl-aetr

Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019)

	Ref	Expression
Assertion	wl-aetr	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 → ∀ 𝑦 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		ax7	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 → 𝑦 = 𝑧 ) )
2	1	al2imi	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )
3		axc11	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑦 = 𝑧 → ∀ 𝑦 𝑦 = 𝑧 ) )
4	2 3	syld	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 → ∀ 𝑦 𝑦 = 𝑧 ) )