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Theorem sbco3v 1962
Description: Version of sbco3 1967 with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbco3v ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbco3v
StepHypRef Expression
1 nfs1v 1932 . . . 4 𝑥[𝑦 / 𝑥]𝜑
21nfri 1512 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
32sbco2vh 1938 . 2 ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4 sbco 1961 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)
54sbbii 1758 . 2 ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
63, 5bitr3i 185 1 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  sbcomv  1964
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