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Theorem elsb3 2123
 Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
Assertion
Ref Expression
elsb3 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ1 2122 . 2 (𝑥 = 𝑤 → (𝑥𝑧𝑤𝑧))
2 elequ1 2122 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
31, 2sbievw2 2108 1 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2070 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 1912  ax-6 1971  ax-7 2016  ax-8 2117 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071 This theorem is referenced by:  cvjust  2819
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