Theorem nfin 4124

Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2180, ax-11 2197, ax-12 2216. (Revised by SN, 14-May-2025.)

Hypotheses

Ref	Expression
nfin.1	⊢ Ⅎ𝑥𝐴
nfin.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfin	⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Proof of Theorem nfin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 4096	. . 3 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		nfin.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfin.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfan 1934	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1886	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∩ 𝐵)
8	7	nfci 2914	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2147  Ⅎwnfc 2911   ∩ cin 4092

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-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-v 3456  df-in 4094

This theorem is used by:  inn0f  4322  csbin  4403  iunxdif3  5060  disjxun  5106  nfres  5988  nfpred  6335  cp  9991  tskwe  10068  iunconn  23801  ptclsg  23989  restmetu  24944  limciun  26269  disjunsn  33286  ordtconnlem1  34667  esum2d  34836  finminlem  37321  bj-rcleqf  38153  mbfposadd  38853  iunconnlem2  46177  disjrnmpt2  46476  disjinfi  46480  fsumiunss  46860  stoweidlem57  47340  fourierdlem80  47469  sge0iunmptlemre  47698  iundjiun  47743  pimiooltgt  47993  smflim  48060  smfpimcclem  48090  smfpimcc  48091  adddmmbl  48116  adddmmbl2  48117  muldmmbl  48118  muldmmbl2  48119  smfdivdmmbl2  48124