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Theorem sbco2v 2272
Description: Version of sbco2 2477 with disjoint variable conditions, not requiring ax-13 2301, but ax-11 2093. (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 2270 . 2 𝑧[𝑦 / 𝑥]𝜑
3 sbequ 2035 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbiev 2253 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wnf 1746  [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  ax-10 2079  ax-11 2093  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-sb 2016
This theorem is referenced by:  ichbi12i  42986
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