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Theorem sbied 1768
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1771). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
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 . . 3 𝑥𝜑
21nfri 1499 . 2 (𝜑 → ∀𝑥𝜑)
3 sbied.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
43nfrd 1500 . 2 (𝜑 → (𝜒 → ∀𝑥𝜒))
5 sbied.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
62, 4, 5sbiedh 1767 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1440  [wsb 1742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743 This theorem is referenced by:  sbiedv  1769  dvelimdf  1996  cbvrald  13321
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