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Theorem sbiedv 1766
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1768). (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 1505 . 2 𝑥𝜑
2 nfvd 1506 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 sbiedv.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 114 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4sbied 1765 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  [wsb 1739 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740 This theorem is referenced by:  acexmid  5813
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