Theorem nfcsbw 3859

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3860 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 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 3833	. . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nftru 1807	. . . 4 ⊢ Ⅎ𝑧⊤
3		nftru 1807	. . . . 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 2896	. . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵)
9	3, 5, 8	nfsbcdw 3737	. . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
10	2, 9	nfabdw 2930	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
11	1, 10	nfcxfrd 2906	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
12	11	mptru 1546	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1540   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  [wsbc 3716  ⦋csb 3832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-sbc 3717  df-csb 3833

This theorem is referenced by:  cbvrabcsfw  3876  elfvmptrab1w  6901  fmptcof  7002  fvmpopr2d  7434  elovmporab1w  7516  mpomptsx  7904  dmmpossx  7906  fmpox  7907  el2mpocsbcl  7925  fmpoco  7935  dfmpo  7942  mpocurryd  8085  fvmpocurryd  8087  nfsum  15402  fsum2dlem  15482  fsumcom2  15486  nfcprod  15621  fprod2dlem  15690  fprodcom2  15694  fsumcn  24033  fsum2cn  24034  dvmptfsum  25139  itgsubst  25213  iundisj2f  30929  f1od2  31056  esumiun  32062  poimirlem26  35803  cdlemkid  38950  cdlemk19x  38957  cdlemk11t  38960  wdom2d2  40857  dmmpossx2  45672