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

Theorem elabf 2680
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabf.1 xψ
elabf.2 A V
elabf.3 (x = A → (φψ))
Assertion
Ref Expression
elabf (A {xφ} ↔ ψ)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem elabf
StepHypRef Expression
1 elabf.2 . 2 A V
2 nfcv 2175 . . 3 xA
3 elabf.1 . . 3 xψ
4 elabf.3 . . 3 (x = A → (φψ))
52, 3, 4elabgf 2679 . 2 (A V → (A {xφ} ↔ ψ))
61, 5ax-mp 7 1 (A {xφ} ↔ ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  Vcvv 2551
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
This theorem is referenced by:  elab  2681  indpi  6326
  Copyright terms: Public domain W3C validator