Theorem nfabd 2359

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 1542	. 2 |- F/ z ph
2		df-clab 2183	. . 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 1996	. . 3 |- ( ph -> F/ x [ z / y ] ps )
6	2, 5	nfxfrd 1489	. 2 |- ( ph -> F/ x z e. { y | ps } )
7	1, 6	nfcd 2334	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -> wi 4   F/wnf 1474   [wsb 1776   e. wcel 2167   {cab 2182   F/_wnfc 2326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-nfc 2328

This theorem is referenced by:  nfsbcd  3009  nfcsb1d  3115  nfcsbd  3120  nfifd  3588  nfunid  3846  nfiotadw  5222  nfixpxy  6776