Theorem nfabd 2395

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 1577	. 2 |- F/ z ph
2		df-clab 2218	. . 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 2030	. . 3 |- ( ph -> F/ x [ z / y ] ps )
6	2, 5	nfxfrd 1524	. 2 |- ( ph -> F/ x z e. { y | ps } )
7	1, 6	nfcd 2370	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -> wi 4   F/wnf 1509   [wsb 1810   e. wcel 2202   {cab 2217   F/_wnfc 2362

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-nfc 2364

This theorem is referenced by:  nfsbcd  3052  nfcsb1d  3159  nfcsbd  3164  nfifd  3637  nfunid  3905  nfiotadw  5296  nfixpxy  6929