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Theorem sbied 2541
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2540) Usage of this theorem is discouraged because it depends on ax-13 2386. See sbiedw 2328, sbiedvw 2100 for variants using disjoint variables, but requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (New usage is discouraged.)
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 2309 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
3 sbied.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
41, 3nfim1 2194 . . . 4 𝑥(𝜑𝜒)
5 sbied.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
65com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
76pm5.74d 275 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
84, 7sbie 2540 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑𝜒))
92, 8bitr3i 279 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑𝜒))
109pm5.74ri 274 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wnf 1780  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by:  sbiedv  2542  sbco2  2549  wl-equsb3  34786
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