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Theorem elsb1 2106
Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2115 for substitution for the second argument. (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
elsb1 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ1 2105 . 2 (𝑥 = 𝑤 → (𝑥𝑧𝑤𝑧))
2 elequ1 2105 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
31, 2sbievw2 2091 1 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060
This theorem is referenced by:  cvjust  2718
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