Theorem nfcsbw 3164

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3165 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfcsbw.1	|- F/_ x A
nfcsbw.2	|- F/_ x B

Assertion

Ref	Expression
nfcsbw	|- F/_ x [_ A / y ]_ B

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem nfcsbw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3128	. . 3 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nftru 1514	. . . 4 |- F/ z T.
3		nftru 1514	. . . . 5 |- F/ y T.
4		nfcsbw.1	. . . . . 6 |- F/_ x A
5	4	a1i 9	. . . . 5 |- ( T. -> F/_ x A )
6		nfcsbw.2	. . . . . . 7 |- F/_ x B
7	6	a1i 9	. . . . . 6 |- ( T. -> F/_ x B )
8	7	nfcrd 2388	. . . . 5 |- ( T. -> F/ x z e. B )
9	3, 5, 8	nfsbcdw 3161	. . . 4 |- ( T. -> F/ x [. A / y ]. z e. B )
10	2, 9	nfabdw 2393	. . 3 |- ( T. -> F/_ x { z | [. A / y ]. z e. B } )
11	1, 10	nfcxfrd 2372	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
12	11	mptru 1406	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1398   e. wcel 2202   {cab 2217   F/_wnfc 2361   [.wsbc 3031   [_csb 3127

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-sbc 3032  df-csb 3128

This theorem is referenced by:  fvmpopr2d  6157  elovmporab1w  6222  fprod2dlemstep  12182  fprodcom2fi  12186  dvmptfsum  15448