Theorem nfsbcdw 3065

Description: Version of nfsbcd 2956 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfsbcdw.1	|- F/ y ph
nfsbcdw.2	|- ( ph -> F/_ x A )
nfsbcdw.3	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfsbcdw	|- ( ph -> F/ x [. A / y ]. ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x, y)

Proof of Theorem nfsbcdw

Step	Hyp	Ref	Expression
1		df-sbc 2938	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
2		nfsbcdw.2	. . 3 |- ( ph -> F/_ x A )
3		nfsbcdw.1	. . . 4 |- F/ y ph
4		nfsbcdw.3	. . . 4 |- ( ph -> F/ x ps )
5	3, 4	nfabdw 2318	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2315	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1455	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1440   e. wcel 2128   {cab 2143   F/_wnfc 2286   [.wsbc 2937

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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938

This theorem is referenced by:  nfcsbw  3067