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Theorem sb9v 1870
Description: Like sb9 1871 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 1830 . 2 ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
2 hbs1 1830 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 sbequ12 1670 . . . 4 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
43equcoms 1610 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
5 sbequ12 1670 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5bitr3d 183 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
71, 2, 6cbvalh 1652 1 (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sb9  1871
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