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Theorem sbeqal1 37517
Description: If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧 is true. (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 2244 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧)
2 equsb3 2324 . 2 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
31, 2sylib 206 1 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  [wsb 1830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1699  df-sb 1831
This theorem is referenced by:  sbeqal1i  37518  sbeqalbi  37520
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