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Theorem elsb4 2128
 Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
Assertion
Ref Expression
elsb4 ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2127 . 2 (𝑥 = 𝑤 → (𝑧𝑥𝑧𝑤))
2 elequ2 2127 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
31, 2sbievw2 2105 1 ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  nfnid  5244
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