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Theorem sbied 1712
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1715). (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 1453 . 2 (𝜑 → ∀𝑥𝜑)
3 sbied.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
43nfrd 1454 . 2 (𝜑 → (𝜒 → ∀𝑥𝜒))
5 sbied.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
62, 4, 5sbiedh 1711 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1390  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sbiedv  1713  dvelimdf  1934  cbvrald  10749
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