Theorem nfin 4183

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 2142, ax-11 2158, ax-12 2178. (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 3927	. . 3 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		nfin.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2883	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfin.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2883	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1853	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∩ 𝐵)
8	7	nfci 2879	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∧ wa 395   ∈ wcel 2109  Ⅎwnfc 2876   ∩ cin 3910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-v 3446  df-in 3918

This theorem is referenced by:  inn0f  4330  csbin  4401  iunxdif3  5054  disjxun  5100  nfres  5941  nfpred  6267  cp  9820  tskwe  9879  iunconn  23291  ptclsg  23478  restmetu  24434  limciun  25771  disjunsn  32496  ordtconnlem1  33887  esum2d  34056  finminlem  36279  bj-rcleqf  36986  mbfposadd  37634  iunconnlem2  44897  disjrnmpt2  45155  disjinfi  45159  fsumiunss  45546  stoweidlem57  46028  fourierdlem80  46157  sge0iunmptlemre  46386  iundjiun  46431  pimiooltgt  46681  smflim  46748  smfpimcclem  46778  smfpimcc  46779  adddmmbl  46804  adddmmbl2  46805  muldmmbl  46806  muldmmbl2  46807  smfdivdmmbl2  46812