Theorem wl-aetr 38212

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

Assertion

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

Proof of Theorem wl-aetr

Step	Hyp	Ref	Expression
1		ax7 2045	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
2	1	al2imi 1844	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
3		axc11 2461	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
4	2, 3	syld 48	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813

This theorem is used by: (None)