Theorem nfcsbw 3925

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3926 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfcsbw.1	⊢ Ⅎ𝑥𝐴
nfcsbw.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfcsbw	⊢ Ⅎ𝑥⦋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 3900	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1804	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1804	. . . . 5 ⊢ Ⅎ𝑦⊤
4		nfcsbw.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . . . 5 ⊢ (⊤ → Ⅎ𝑥𝐴)
6		nfcsbw.2	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐵)
8	7	nfcrd 2899	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3809	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2927	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2904	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1547	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1541   ∈ wcel 2108  {cab 2714  Ⅎwnfc 2890  [wsbc 3788  ⦋csb 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-sbc 3789  df-csb 3900

This theorem is referenced by:  cbvrabcsfw  3940  elfvmptrab1w  7043  fmptcof  7150  fvmpopr2d  7595  elovmporab1w  7680  mpomptsx  8089  dmmpossx  8091  fmpox  8092  el2mpocsbcl  8110  fmpoco  8120  dfmpo  8127  mpocurryd  8294  fvmpocurryd  8296  nfsum  15727  fsum2dlem  15806  fsumcom2  15810  nfcprod  15945  fprod2dlem  16016  fprodcom2  16020  fsumcn  24894  fsum2cn  24895  dvmptfsum  26013  itgsubst  26090  iundisj2f  32603  f1od2  32732  esumiun  34095  poimirlem26  37653  cdlemkid  40938  cdlemk19x  40945  cdlemk11t  40948  fmpocos  42275  wdom2d2  43047  dmmpossx2  48253