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Theorem sbiedv 2546
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2544). Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker sbiedvw 2104 when possible. (Contributed by NM, 7-Jan-2017.) (New usage is discouraged.)
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 1915 . 2 𝑥𝜑
2 nfvd 1916 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 sbiedv.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 416 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4sbied 2545 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  2sbiev  2547
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