Theorem nfcsbw 3871

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3872 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2372. (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 3846	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1805	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1805	. . . . 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 2888	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3757	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2916	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2893	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1548	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1542   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  [wsbc 3736  ⦋csb 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-sbc 3737  df-csb 3846

This theorem is referenced by:  cbvrabcsfw  3886  elfvmptrab1w  6962  fmptcof  7069  fvmpopr2d  7514  elovmporab1w  7599  mpomptsx  8002  dmmpossx  8004  fmpox  8005  el2mpocsbcl  8021  fmpoco  8031  dfmpo  8038  mpocurryd  8205  fvmpocurryd  8207  nfsum  15604  fsum2dlem  15683  fsumcom2  15687  nfcprod  15822  fprod2dlem  15893  fprodcom2  15897  fsumcn  24794  fsum2cn  24795  dvmptfsum  25912  itgsubst  25989  iundisj2f  32577  f1od2  32709  esumiun  34114  poimirlem26  37692  cdlemkid  41041  cdlemk19x  41048  cdlemk11t  41051  fmpocos  42333  wdom2d2  43133  dmmpossx2  48442