ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcied2 GIF version

Theorem sbcied2 2949
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1 (𝜑𝐴𝑉)
sbcied2.2 (𝜑𝐴 = 𝐵)
sbcied2.3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
sbcied2 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2 (𝜑𝐴𝑉)
2 id 19 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 sbcied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2195 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 sbcied2.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
64, 5syldan 280 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
71, 6sbcied 2948 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  [wsbc 2912
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator