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

Theorem sbcex 2766
 Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcex ([A / x]φA V)

Proof of Theorem sbcex
StepHypRef Expression
1 df-sbc 2759 . 2 ([A / x]φA {xφ})
2 elex 2560 . 2 (A {xφ} → A V)
31, 2sylbi 114 1 ([A / x]φA V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  {cab 2023  Vcvv 2551  [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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-sbc 2759 This theorem is referenced by:  sbcco  2779  sbc5  2781  sbcan  2799  sbcor  2801  sbcal  2804  sbcex2  2806  spesbc  2837  csbprc  3256  opelopabsb  3988
 Copyright terms: Public domain W3C validator