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Theorem equsb3r 2103
Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2102. (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 2024 . 2 (𝑥 = 𝑤 → (𝑧 = 𝑥𝑧 = 𝑤))
2 equequ2 2024 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
31, 2sbievw2 2097 1 ([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064
This theorem is referenced by:  icheq  47454
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