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Theorem sb9v 1978
Description: Like sb9 1979 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 1938 . 2 ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
2 hbs1 1938 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 sbequ12 1771 . . . 4 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
43equcoms 1708 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
5 sbequ12 1771 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5bitr3d 190 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
71, 2, 6cbvalh 1753 1 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  sb9  1979
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