Theorem nfabd 2319

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 1508	. 2 |- F/ z ph
2		df-clab 2144	. . 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 1957	. . 3 |- ( ph -> F/ x [ z / y ] ps )
6	2, 5	nfxfrd 1455	. 2 |- ( ph -> F/ x z e. { y | ps } )
7	1, 6	nfcd 2294	1 |- ( ph -> F/_ x { y | ps } )

Syntax hints:   -> wi 4   F/wnf 1440   [wsb 1742   e. wcel 2128   {cab 2143   F/_wnfc 2286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-nfc 2288

This theorem is referenced by:  nfsbcd  2956  nfcsb1d  3062  nfcsbd  3066  nfifd  3532  nfunid  3779  nfiotadw  5138  nfixpxy  6662