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Theorem sbco2v 1928
 Description: Version of sbco2 1945 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
Hypothesis
Ref Expression
sbco2v.1 𝑧𝜑
Assertion
Ref Expression
sbco2v ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbco2v
StepHypRef Expression
1 sbco2v.1 . . 3 𝑧𝜑
21nfsbv 1927 . 2 𝑧[𝑦 / 𝑥]𝜑
3 sbequ 1820 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbiev 1772 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743 This theorem is referenced by: (None)
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