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Theorem equsb3lem 2564
Description: Lemma for equsb3 2565. (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 1988 . 2 𝑦 𝑥 = 𝑧
2 equequ1 2103 . 2 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
31, 2sbie 2541 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 2042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-10 2164  ax-12 2192  ax-13 2387
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1850  df-nf 1855  df-sb 2043
This theorem is referenced by:  equsb3  2565  equsb3ALT  2566
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