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Theorem sbiedv 1730
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1732). (Contributed by NM, 7-Jan-2017.)
Hypothesis
Ref Expression
sbiedv.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
sbiedv (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem sbiedv
StepHypRef Expression
1 nfv 1476 . 2 𝑥𝜑
2 nfvd 1477 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 sbiedv.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 114 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4sbied 1729 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  [wsb 1703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704
This theorem is referenced by:  acexmid  5705
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