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Theorem sbeqal1 40607
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 2497 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧)
2 equsb3 2100 . 2 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
31, 2sylib 219 1 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  sbeqal1i  40608
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