Theorem nfin 4175

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 3919	. . 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 3902

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 3438  df-in 3910

This theorem is referenced by:  inn0f  4322  csbin  4393  iunxdif3  5044  disjxun  5090  nfres  5932  nfpred  6254  cp  9787  tskwe  9846  iunconn  23313  ptclsg  23500  restmetu  24456  limciun  25793  disjunsn  32538  ordtconnlem1  33891  esum2d  34060  finminlem  36292  bj-rcleqf  36999  mbfposadd  37647  iunconnlem2  44908  disjrnmpt2  45166  disjinfi  45170  fsumiunss  45556  stoweidlem57  46038  fourierdlem80  46167  sge0iunmptlemre  46396  iundjiun  46441  pimiooltgt  46691  smflim  46758  smfpimcclem  46788  smfpimcc  46789  adddmmbl  46814  adddmmbl2  46815  muldmmbl  46816  muldmmbl2  46817  smfdivdmmbl2  46822