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Theorem eqsb3lem 2197
 Description: Lemma for eqsb3 2198. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
eqsb3lem ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3lem
StepHypRef Expression
1 nfv 1473 . 2 𝑦 𝑥 = 𝐴
2 eqeq1 2101 . 2 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
31, 2sbie 1728 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1296  [wsb 1699 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 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-cleq 2088 This theorem is referenced by:  eqsb3  2198
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