Theorem wl-ax11-lem1

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

	Ref	Expression
Assertion	wl-ax11-lem1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 ↔ ∀ 𝑦 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		wl-aetr	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 → ∀ 𝑦 𝑦 = 𝑧 ) )
2		wl-aetr	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑦 𝑦 = 𝑧 → ∀ 𝑥 𝑥 = 𝑧 ) )
3	2	aecoms	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝑦 = 𝑧 → ∀ 𝑥 𝑥 = 𝑧 ) )
4	1 3	impbid	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑧 ↔ ∀ 𝑦 𝑦 = 𝑧 ) )