MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfopab1 Unicode version

Theorem nfopab1 4266
Description: The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
nfopab1  |-  F/_ x { <. x ,  y
>.  |  ph }

Proof of Theorem nfopab1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 4259 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1747 . . 3  |-  F/ x E. x E. y ( z  =  <. x ,  y >.  /\  ph )
32nfab 2575 . 2  |-  F/_ x { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
41, 3nfcxfr 2568 1  |-  F/_ x { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652   {cab 2421   F/_wnfc 2558   <.cop 3809   {copab 4257
This theorem is referenced by:  nfmpt1  4290  opelopabsb  4457  ssopab2b  4473  dmopab  5071  rnopab  5106  funopab  5477  zfrep6  5959  0neqopab  6110  fvopab5  6525  aomclem8  27074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-opab 4259
  Copyright terms: Public domain W3C validator