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Theorem sbiedw 2323
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2322). Version of sbied 2522 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2142. (Revised by Wolf Lammen, 28-Jan-2024.)
Hypotheses
Ref Expression
sbiedw.1 𝑥𝜑
sbiedw.2 (𝜑 → Ⅎ𝑥𝜒)
sbiedw.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
sbiedw (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem sbiedw
StepHypRef Expression
1 sbiedw.1 . . . 4 𝑥𝜑
21sbrimv 2310 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
3 sbiedw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
41, 3nfim1 2197 . . . 4 𝑥(𝜑𝜒)
5 sbiedw.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
65com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
76pm5.74d 276 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
84, 7sbiev 2322 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑𝜒))
92, 8bitr3i 280 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑𝜒))
109pm5.74ri 275 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  csbie2df  4337
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