Theorem nfsbcdw 3079

Description: Version of nfsbcd 2970 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 2952	. 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 2327	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2324	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1463	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   [.wsbc 2951

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952

This theorem is referenced by:  nfcsbw  3081