Theorem nfabd2 2784

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.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ z ( ph /\ -. A. x x = y )
2		df-clab 2609	. . . . 5 |- ( z e. { y | ps } <-> [ z / y ] ps )
3		nfabd2.1	. . . . . . 7 |- F/ y ph
4		nfnae 2318	. . . . . . 7 |- F/ y -. A. x x = y
5	3, 4	nfan 1828	. . . . . 6 |- F/ y ( ph /\ -. A. x x = y )
6		nfabd2.2	. . . . . 6 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
7	5, 6	nfsbd 2442	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/ x [ z / y ] ps )
8	2, 7	nfxfrd 1780	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x z e. { y | ps } )
9	1, 8	nfcd 2759	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/_ x { y | ps } )
10	9	ex 450	. 2 |- ( ph -> ( -. A. x x = y -> F/_ x { y | ps } ) )
11		nfab1 2766	. . 3 |- F/_ y { y | ps }
12		eqidd 2623	. . . 4 |- ( A. x x = y -> { y | ps } = { y | ps } )
13	12	drnfc1 2782	. . 3 |- ( A. x x = y -> ( F/_ x { y | ps } <-> F/_ y { y | ps } ) )
14	11, 13	mpbiri 248	. 2 |- ( A. x x = y -> F/_ x { y | ps } )
15	10, 14	pm2.61d2 172	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   F/wnf 1708   [wsb 1880   e. wcel 1990   {cab 2608   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfabd  2785  nfrab  3123  nfixp  7927