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Theorem sbeqal1 39536
 Description: If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.)
Assertion
Ref Expression
sbeqal1 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem sbeqal1
StepHypRef Expression
1 sb2 2426 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧)
2 equsb3 2509 . 2 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
31, 2sylib 210 1 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1599  [wsb 2011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162  ax-13 2333 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012 This theorem is referenced by:  sbeqal1i  39537
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