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Theorem sbcov 2251
Description: Version of sbco 2547 with a disjoint variable condition using fewer axioms. (Contributed by Gino Giotto, 7-Aug-2023.)
Assertion
Ref Expression
sbcov ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbcov
StepHypRef Expression
1 sbcom3vv 2099 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2 sbid 2250 . . 3 ([𝑦 / 𝑦]𝜑𝜑)
32sbbii 2074 . 2 ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitri 276 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2062
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 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063
This theorem is referenced by:  sb6a  2252  sbbibOLD  2283
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