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Theorem nfabd2 2588
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfabd2.1  |-  F/ y
ph
nfabd2.2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
Assertion
Ref Expression
nfabd2  |-  ( ph  -> 
F/_ x { y  |  ps } )

Proof of Theorem nfabd2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1755 . . . 4  |-  F/ z ( ph  /\  -.  A. x  x  =  y )
2 df-clab 2415 . . . . 5  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
3 nfabd2.1 . . . . . . 7  |-  F/ y
ph
4 nfnae 2124 . . . . . . 7  |-  F/ y  -.  A. x  x  =  y
53, 4nfan 1988 . . . . . 6  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
6 nfabd2.2 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
75, 6nfsbd 2248 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x [ z  /  y ] ps )
82, 7nfxfrd 1691 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  z  e. 
{ y  |  ps } )
91, 8nfcd 2564 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x { y  |  ps } )
109ex 435 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x { y  |  ps } ) )
11 nfab1 2571 . . 3  |-  F/_ y { y  |  ps }
12 eqidd 2429 . . . 4  |-  ( A. x  x  =  y  ->  { y  |  ps }  =  { y  |  ps } )
1312drnfc1 2586 . . 3  |-  ( A. x  x  =  y  ->  ( F/_ x {
y  |  ps }  <->  F/_ y { y  |  ps } ) )
1411, 13mpbiri 236 . 2  |-  ( A. x  x  =  y  -> 
F/_ x { y  |  ps } )
1510, 14pm2.61d2 163 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   F/wnf 1661   [wsb 1790    e. wcel 1872   {cab 2414   F/_wnfc 2556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558
This theorem is referenced by:  nfabd  2589  nfrab  2949  nfixp  7496
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