Theorem nfcsbw 3175

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3176 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 3139	. . 3 |- [_ A / y ]_ B = { z | [. A / y ]. z e. B }
2		nftru 1515	. . . 4 |- F/ z T.
3		nftru 1515	. . . . 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 2398	. . . . 5 |- ( T. -> F/ x z e. B )
9	3, 5, 8	nfsbcdw 3172	. . . 4 |- ( T. -> F/ x [. A / y ]. z e. B )
10	2, 9	nfabdw 2403	. . 3 |- ( T. -> F/_ x { z | [. A / y ]. z e. B } )
11	1, 10	nfcxfrd 2382	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
12	11	mptru 1407	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1399   e. wcel 2203   {cab 2218   F/_wnfc 2371   [.wsbc 3042   [_csb 3138

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-sbc 3043  df-csb 3139

This theorem is referenced by:  fvmpopr2d  6190  elovmporab1w  6255  fprod2dlemstep  12308  fprodcom2fi  12312  dvmptfsum  15590