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Theorem equsb3lem 2419
 Description: Lemma for equsb3 2420. (Contributed by Raph Levien and FL, 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 nfv 1830 . 2 𝑦 𝑥 = 𝑧
2 equequ1 1939 . 2 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
31, 2sbie 2396 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  equsb3  2420  equsb3ALT  2421
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