Theorem nfsbcdw 3135

Description: Version of nfsbcd 3025 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 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 3006	. 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 2369	. . 3 |- ( ph -> F/_ x { y | ps } )
6	2, 5	nfeld 2366	. 2 |- ( ph -> F/ x A e. { y | ps } )
7	1, 6	nfxfrd 1499	1 |- ( ph -> F/ x [. A / y ]. ps )

Syntax hints:   -> wi 4   F/wnf 1484   e. wcel 2178   {cab 2193   F/_wnfc 2337   [.wsbc 3005

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-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-sbc 3006

This theorem is referenced by:  nfsbcw  3136  nfcsbw  3138