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

Theorem nfopab1 4085
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 4078 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 nfe1 1706 . . 3  |-  F/ x E. x E. y ( z  =  <. x ,  y >.  /\  ph )
32nfab 2423 . 2  |-  F/_ x { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
41, 3nfcxfr 2416 1  |-  F/_ x { <. x ,  y
>.  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623   {cab 2269   F/_wnfc 2406   <.cop 3643   {copab 4076
This theorem is referenced by:  nfmpt1  4109  opelopabsb  4275  ssopab2b  4291  dmopab  4889  rnopab  4924  funopab  5287  zfrep6  5748  fvopab5  6289  aomclem8  27159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-opab 4078
  Copyright terms: Public domain W3C validator