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Theorem eqsb3lem 2876
Description: Lemma for eqsb3 2877. (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 1995 . 2 𝑦 𝑥 = 𝐴
2 eqeq1 2775 . 2 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
31, 2sbie 2555 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050  df-cleq 2764
This theorem is referenced by:  eqsb3  2877
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