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Theorem sbied 2397
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2396). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)
Hypotheses
Ref Expression
sbied.1 𝑥𝜑
sbied.2 (𝜑 → Ⅎ𝑥𝜒)
sbied.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
sbied (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Proof of Theorem sbied
StepHypRef Expression
1 sbied.1 . . . 4 𝑥𝜑
21sbrim 2384 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
3 sbied.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
41, 3nfim1 2055 . . . 4 𝑥(𝜑𝜒)
5 sbied.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
65com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
76pm5.74d 261 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
84, 7sbie 2396 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑𝜒))
92, 8bitr3i 265 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑𝜒))
109pm5.74ri 260 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  sbiedv  2398  sbco2  2403  wl-equsb3  32516
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