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Theorem sbiev 1772
 Description: Conversion of implicit substitution to explicit substitution. Version of sbie 1771 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
Hypotheses
Ref Expression
sbiev.1 𝑥𝜓
sbiev.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbiev ([𝑦 / 𝑥]𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbiev
StepHypRef Expression
1 sbiev.1 . 2 𝑥𝜓
2 sbiev.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2sbie 1771 1 ([𝑦 / 𝑥]𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1440  [wsb 1742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743 This theorem is referenced by:  sbco2v  1928  cbvabw  2280  csbcow  3042
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