MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbiedv Structured version   Visualization version   GIF version

Theorem sbiedv 2409
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2407). (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 1840 . 2 𝑥𝜑
2 nfvd 1841 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 sbiedv.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 450 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4sbied 2408 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  2mos  2551  iscatd2  16270  prtlem5  33651
  Copyright terms: Public domain W3C validator