 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbc7 Structured version   GIF version

Theorem sbc7 2784
 Description: An equivalence for class substitution in the spirit of df-clab 2024. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbc7 ([A / x]φy(y = A [y / x]φ))
Distinct variable groups:   y,A   φ,y   x,y
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem sbc7
StepHypRef Expression
1 sbcco 2779 . 2 ([A / y][y / x]φ[A / x]φ)
2 sbc5 2781 . 2 ([A / y][y / x]φy(y = A [y / x]φ))
31, 2bitr3i 175 1 ([A / x]φy(y = A [y / x]φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378  [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-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: (None)
 Copyright terms: Public domain W3C validator