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Theorem sbco2v 1958
Description: Version of sbco2 1975 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 1957 . 2 𝑧[𝑦 / 𝑥]𝜑
3 sbequ 1850 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbiev 1802 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 105  wnf 1470  [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by: (None)
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