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Theorem sbeqal1i 38916
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 38915 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
2 sbeqal1i.1 . 2 (𝑥 = 𝑦𝑥 = 𝑧)
31, 2mpg 1764 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 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbeqal2i  38917
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