Theorem nfcsbw 3877

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3878 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2404. (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 3852	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1837	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1837	. . . . 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 2920	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3763	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2947	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2925	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1577	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

This proof depends on syntax axioms:  ⊤wtru 1571   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911  [wsbc 3742  ⦋csb 3851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-sbc 3743  df-csb 3852

This theorem is used by:  cbvrabcsfw  3892  elfvmptrab1w  7044  fmptcof  7155  fvmpopr2d  7606  elovmporab1w  7692  mpomptsx  8100  dmmpossx  8102  fmpox  8103  el2mpocsbcl  8121  fmpoco  8132  f1od2  8135  dfmpo  8140  mpocurryd  8314  fvmpocurryd  8316  nfsum  15887  fsum2dlem  15965  fsumcom2  15969  nfcprod  16107  fprod2dlem  16176  fprodcom2  16180  fsumcn  25221  fsum2cn  25222  dvmptfsum  26325  itgsubst  26399  iundisj2f  33256  esumiun  34796  poimirlem26  38756  cdlemkid  42185  cdlemk19x  42192  cdlemk11t  42195  fmpocos  43480  wdom2d2  44235  dmmpossx2  49654