Theorem nfcsbw 3872

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3873 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2401. (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 3847	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1836	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1836	. . . . 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 2916	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3759	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2943	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2921	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1576	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

This proof depends on syntax axioms:  ⊤wtru 1570   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-sbc 3739  df-csb 3847

This theorem is used by:  cbvrabcsfw  3887  elfvmptrab1w  7021  fmptcof  7131  fvmpopr2d  7582  elovmporab1w  7668  mpomptsx  8075  dmmpossx  8077  fmpox  8078  el2mpocsbcl  8096  fmpoco  8106  dfmpo  8113  mpocurryd  8286  fvmpocurryd  8288  nfsum  15858  fsum2dlem  15936  fsumcom2  15940  nfcprod  16078  fprod2dlem  16147  fprodcom2  16151  fsumcn  25191  fsum2cn  25192  dvmptfsum  26295  itgsubst  26369  iundisj2f  33226  f1od2  33353  esumiun  34768  poimirlem26  38607  cdlemkid  42036  cdlemk19x  42043  cdlemk11t  42046  fmpocos  43330  wdom2d2  44085  dmmpossx2  49492