Theorem wl-aetr 35688

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 2019	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
2	1	al2imi 1818	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
3		axc11 2430	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
4	2, 3	syld 47	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by:  wl-ax11-lem1  35736  wl-ax11-lem3  35738