Theorem nfcsbw 3876

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3877 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 3851	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1806	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1806	. . . . 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 2893	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3762	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2921	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2898	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1549	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1543   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884  [wsbc 3741  ⦋csb 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-sbc 3742  df-csb 3851

This theorem is referenced by:  cbvrabcsfw  3891  elfvmptrab1w  6970  fmptcof  7078  fvmpopr2d  7523  elovmporab1w  7608  mpomptsx  8011  dmmpossx  8013  fmpox  8014  el2mpocsbcl  8030  fmpoco  8040  dfmpo  8047  mpocurryd  8214  fvmpocurryd  8216  nfsum  15619  fsum2dlem  15698  fsumcom2  15702  nfcprod  15837  fprod2dlem  15908  fprodcom2  15912  fsumcn  24822  fsum2cn  24823  dvmptfsum  25940  itgsubst  26017  iundisj2f  32669  f1od2  32801  esumiun  34264  poimirlem26  37860  cdlemkid  41275  cdlemk19x  41282  cdlemk11t  41285  fmpocos  42569  wdom2d2  43355  dmmpossx2  48660