Theorem nfcsbw 3900

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3901 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2376. (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 3875	. . 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 2892	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3786	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2920	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2897	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1547	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1541   ∈ wcel 2108  {cab 2713  Ⅎwnfc 2883  [wsbc 3765  ⦋csb 3874

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-sbc 3766  df-csb 3875

This theorem is referenced by:  cbvrabcsfw  3915  elfvmptrab1w  7012  fmptcof  7119  fvmpopr2d  7567  elovmporab1w  7652  mpomptsx  8061  dmmpossx  8063  fmpox  8064  el2mpocsbcl  8082  fmpoco  8092  dfmpo  8099  mpocurryd  8266  fvmpocurryd  8268  nfsum  15705  fsum2dlem  15784  fsumcom2  15788  nfcprod  15923  fprod2dlem  15994  fprodcom2  15998  fsumcn  24810  fsum2cn  24811  dvmptfsum  25929  itgsubst  26006  iundisj2f  32517  f1od2  32644  esumiun  34071  poimirlem26  37616  cdlemkid  40901  cdlemk19x  40908  cdlemk11t  40911  fmpocos  42232  wdom2d2  43006  dmmpossx2  48260