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Theorem sblimv 1774
 Description: Version of sblim 1831 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
Hypothesis
Ref Expression
sblimv.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
sblimv ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sblimv
StepHypRef Expression
1 sbimv 1773 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sblimv.1 . . . 4 (𝜓 → ∀𝑥𝜓)
32sbh 1659 . . 3 ([𝑦 / 𝑥]𝜓𝜓)
43imbi2i 215 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
51, 4bitri 173 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by: (None)
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