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Theorem sbco2vv 2044
Description: Version of sbco2 2477 with disjoint variable conditions and fewer axioms. (Contributed by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.)
Assertion
Ref Expression
sbco2vv ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbco2vv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbequ 2035 . 2 (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
2 sbequ 2035 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2sbievw2 2043 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  [wsb 2015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016
This theorem is referenced by:  sbid2vw  2186  sbco4lem  2412  sbco4  2413  clelsb3vOLD  2895  cbvabv  2911  sbralie  3398  wl-equsb3  34230  wl-dfrabv  34300  wl-dfrabf  34302  2reu8i  42716
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