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Theorem sb9v 1953
Description: Like sb9 1954 but with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 28-Feb-2018.)
Assertion
Ref Expression
sb9v (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb9v
StepHypRef Expression
1 hbs1 1911 . 2 ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
2 hbs1 1911 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 sbequ12 1744 . . . 4 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
43equcoms 1684 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
5 sbequ12 1744 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5bitr3d 189 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
71, 2, 6cbvalh 1726 1 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1329  [wsb 1735
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sb9  1954
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