Theorem nfcsbw 3878

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3879 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2402. (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 3853	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1832	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1832	. . . . 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 2917	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3764	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2944	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2922	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1575	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1569   ∈ wcel 2141  {cab 2739  Ⅎwnfc 2908  [wsbc 3743  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-sbc 3744  df-csb 3853

This theorem is referenced by:  cbvrabcsfw  3893  elfvmptrab1w  7017  fmptcof  7126  fvmpopr2d  7572  elovmporab1w  7657  mpomptsx  8060  dmmpossx  8062  fmpox  8063  el2mpocsbcl  8079  fmpoco  8089  dfmpo  8096  mpocurryd  8264  fvmpocurryd  8266  nfsum  15741  fsum2dlem  15820  fsumcom2  15824  nfcprod  15962  fprod2dlem  16033  fprodcom2  16037  fsumcn  25008  fsum2cn  25009  dvmptfsum  26113  itgsubst  26187  iundisj2f  32901  f1od2  33030  esumiun  34450  poimirlem26  38241  cdlemkid  41656  cdlemk19x  41663  cdlemk11t  41666  fmpocos  42950  wdom2d2  43710  dmmpossx2  49062