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Theorem sbeqal1i 44395
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 44394 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
2 sbeqal1i.1 . 2 (𝑥 = 𝑦𝑥 = 𝑧)
31, 2mpg 1794 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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  sbeqal2i  44396
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