Theorem nfabd 2196

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

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ z ph
2		df-clab 2027	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
3		nfabd.1	. . . 4 |- F/ y ph
4		nfabd.2	. . . 4 |- ( ph -> F/ x ps )
5	3, 4	nfsbd 1851	. . 3 |- ( ph -> F/ x [ z / y ] ps )
6	2, 5	nfxfrd 1364	. 2 |- ( ph -> F/ x z e. { y | ps } )
7	1, 6	nfcd 2173	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -> wi 4   F/wnf 1349   e. wcel 1393   [wsb 1645   {cab 2026   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-nfc 2167

This theorem is referenced by:  nfsbcd  2783  nfcsb1d  2880  nfcsbd  2883  nfifd  3355  nfunid  3587  nfiotadxy  4870