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Theorem equsb3lem 1873
Description: Lemma for equsb3 1874. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
equsb3lem ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable groups:   𝑦,𝑧   𝑥,𝑦

Proof of Theorem equsb3lem
StepHypRef Expression
1 ax-17 1465 . 2 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
2 equequ1 1646 . 2 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
31, 2sbieh 1721 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1290  [wsb 1693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473
This theorem depends on definitions:  df-bi 116  df-sb 1694
This theorem is referenced by:  equsb3  1874
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