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Theorem hbs1 1862
Description: 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)
Assertion
Ref Expression
hbs1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem hbs1
StepHypRef Expression
1 sb6 1814 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 ax-ial 1472 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
31, 2hbxfrbi 1406 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  nfs1v  1863  sb9v  1902  eu1  1973  mopick  2026  hbab1  2077
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