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

Theorem elrab 2692
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
Hypothesis
Ref Expression
elrab.1 (x = A → (φψ))
Assertion
Ref Expression
elrab (A {x Bφ} ↔ (A B ψ))
Distinct variable groups:   ψ,x   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem elrab
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfcv 2175 . 2 xB
3 nfv 1418 . 2 xψ
4 elrab.1 . 2 (x = A → (φψ))
51, 2, 3, 4elrabf 2690 1 (A {x Bφ} ↔ (A B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {crab 2304
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-rab 2309  df-v 2553
This theorem is referenced by:  elrab3  2693  elrab2  2694  ralrab  2696  rexrab  2698  rabsnt  3436  unimax  3605  ssintub  3624  intminss  3631  rabxfrd  4167  onsucelsucexmidlem1  4213  sefvex  5139  ssimaex  5177  acexmidlem2  5452  ssfiexmid  6254  nnind  7691  peano2uz2  8101  peano5uzti  8102  dfuzi  8104  uzind  8105  uzind3  8107  eluz1  8233  uzind4  8287  eqreznegel  8305  elixx1  8516  elioo2  8540  elfz1  8629  expcl2lemap  8901  expclzaplem  8913  expclzap  8914  expap0i  8921  expge0  8925  expge1  8926
  Copyright terms: Public domain W3C validator