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Theorem nfabd2 2565
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 1715 . . . 4  |-  F/ z ( ph  /\  -.  A. x  x  =  y )
2 df-clab 2368 . . . . 5  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
3 nfabd2.1 . . . . . . 7  |-  F/ y
ph
4 nfnae 2064 . . . . . . 7  |-  F/ y  -.  A. x  x  =  y
53, 4nfan 1936 . . . . . 6  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
6 nfabd2.2 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
75, 6nfsbd 2190 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x [ z  /  y ] ps )
82, 7nfxfrd 1654 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  z  e. 
{ y  |  ps } )
91, 8nfcd 2538 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x { y  |  ps } )
109ex 432 . 2  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x { y  |  ps } ) )
11 nfab1 2546 . . 3  |-  F/_ y { y  |  ps }
12 eqidd 2383 . . . 4  |-  ( A. x  x  =  y  ->  { y  |  ps }  =  { y  |  ps } )
1312drnfc1 2563 . . 3  |-  ( A. x  x  =  y  ->  ( F/_ x {
y  |  ps }  <->  F/_ y { y  |  ps } ) )
1411, 13mpbiri 233 . 2  |-  ( A. x  x  =  y  -> 
F/_ x { y  |  ps } )
1510, 14pm2.61d2 160 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1397   F/wnf 1624   [wsb 1747    e. wcel 1826   {cab 2367   F/_wnfc 2530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532
This theorem is referenced by:  nfabd  2566  nfrab  2964  nfixp  7407
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