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Theorem wl-ax11-lem1 34374
 Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))

Proof of Theorem wl-ax11-lem1
StepHypRef Expression
1 wl-aetr 34328 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
2 wl-aetr 34328 . . 3 (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
32aecoms 2407 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
41, 3impbid 213 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-nf 1766 This theorem is referenced by:  wl-ax11-lem8  34381
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