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Theorem sbeqal1i 37421
Description: Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.)
Hypothesis
Ref Expression
sbeqal1i.1 (𝑥 = 𝑦𝑥 = 𝑧)
Assertion
Ref Expression
sbeqal1i 𝑦 = 𝑧
Distinct variable group:   𝑥,𝑧

Proof of Theorem sbeqal1i
StepHypRef Expression
1 sbeqal1 37420 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
2 sbeqal1i.1 . 2 (𝑥 = 𝑦𝑥 = 𝑧)
31, 2mpg 1714 1 𝑦 = 𝑧
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  sbeqal2i  37422
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