Theorem nfcsbw 3874

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3875 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2371. (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 3849	. . 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 2886	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3760	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2914	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2891	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1548	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1542   ∈ wcel 2110  {cab 2708  Ⅎwnfc 2877  [wsbc 3739  ⦋csb 3848

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

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 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-sbc 3740  df-csb 3849

This theorem is referenced by:  cbvrabcsfw  3889  elfvmptrab1w  6951  fmptcof  7058  fvmpopr2d  7503  elovmporab1w  7588  mpomptsx  7991  dmmpossx  7993  fmpox  7994  el2mpocsbcl  8010  fmpoco  8020  dfmpo  8027  mpocurryd  8194  fvmpocurryd  8196  nfsum  15590  fsum2dlem  15669  fsumcom2  15673  nfcprod  15808  fprod2dlem  15879  fprodcom2  15883  fsumcn  24781  fsum2cn  24782  dvmptfsum  25899  itgsubst  25976  iundisj2f  32560  f1od2  32692  esumiun  34097  poimirlem26  37665  cdlemkid  40954  cdlemk19x  40961  cdlemk11t  40964  fmpocos  42246  wdom2d2  43047  dmmpossx2  48347