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Theorem equsb3r 2108
 Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2107. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)
Assertion
Ref Expression
equsb3r ([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem equsb3r
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equequ2 2034 . 2 (𝑥 = 𝑤 → (𝑧 = 𝑥𝑧 = 𝑤))
2 equequ2 2034 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
31, 2sbievw2 2105 1 ([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
 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 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-sb 2071 This theorem is referenced by:  icheq  44348
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