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Theorem sbco2vv 2092
Description: A composition law for substitution. Version of sbco2 2502 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised 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 2078 . 2 (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
2 sbequ 2078 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2sbievw2 2091 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060
This theorem is referenced by:  sbco4lem  2264  sbco4lemOLD  2265  sbco4  2266  cbvsbv  2351  sbralie  3346  sbralieALT  3347  sbccow  3792  wl-equsb3  36911  2reu8i  46306
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