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

Theorem sbcied 2793
 Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1 (φA 𝑉)
sbcied.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
sbcied (φ → ([A / x]ψχ))
Distinct variable groups:   x,A   φ,x   χ,x
Allowed substitution hints:   ψ(x)   𝑉(x)

Proof of Theorem sbcied
StepHypRef Expression
1 sbcied.1 . 2 (φA 𝑉)
2 sbcied.2 . 2 ((φ x = A) → (ψχ))
3 nfv 1418 . 2 xφ
4 nfvd 1419 . 2 (φ → Ⅎxχ)
51, 2, 3, 4sbciedf 2792 1 (φ → ([A / x]ψχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsbc 2758 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759 This theorem is referenced by:  sbcied2  2794  sbc2iedv  2824  sbc3ie  2825  sbcralt  2828  sbcrext  2829  euotd  3982  riota5f  5435
 Copyright terms: Public domain W3C validator