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Theorem sbco2v 2344
Description: A composition law for substitution. Version of sbco2 2533 with disjoint variable conditions, not requiring ax-13 2382, but ax-11 2159. (Contributed by NM, 30-Jun-1994.) (Revised 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 2341 . 2 𝑧[𝑦 / 𝑥]𝜑
3 sbequ 2089 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbiev 2324 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wnf 1785  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  cbvabwOLD  2871  clelsb3fw  2962  ichbi12i  43974
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