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

Theorem nfabd 2451
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfabd.1  |-  F/ y
ph
nfabd.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfabd  |-  ( ph  -> 
F/_ x { y  |  ps } )

Proof of Theorem nfabd
StepHypRef Expression
1 nfabd.1 . 2  |-  F/ y
ph
2 nfabd.2 . . 3  |-  ( ph  ->  F/ x ps )
32adantr 451 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
41, 3nfabd2 2450 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   F/wnf 1534    = wceq 1632   {cab 2282   F/_wnfc 2419
This theorem is referenced by:  nfsbcd  3024  nfcsb1d  3124  nfcsbd  3127  nfifd  3601  nfunid  3850  nfiotad  5238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421
  Copyright terms: Public domain W3C validator