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Theorem wl-aetr 34927
 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
StepHypRef Expression
1 ax7 2023 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21al2imi 1817 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
3 axc11 2444 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
42, 3syld 47 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  wl-ax11-lem1  34975  wl-ax11-lem3  34977
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