Theorem nfabd 2412

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 1581	. 2 |- F/ z ph
2		df-clab 2225	. . 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 2037	. . 3 |- ( ph -> F/ x [ z / y ] ps )
6	2, 5	nfxfrd 1528	. 2 |- ( ph -> F/ x z e. { y | ps } )
7	1, 6	nfcd 2387	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -> wi 4   F/wnf 1513   [wsb 1815   e. wcel 2209   {cab 2224   F/_wnfc 2379

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-nfc 2381

This theorem is referenced by:  nfsbcd  3071  nfcsb1d  3178  nfcsbd  3183  nfifd  3665  nfunid  3937  nfiotadw  5335  nfixpxy  6989