Theorem nfin 4217

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 2130, ax-11 2147, ax-12 2167. (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 3963	. . 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 1895	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1848	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∩ 𝐵)
8	7	nfci 2879	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∧ wa 394   ∈ wcel 2099  Ⅎwnfc 2876   ∩ cin 3946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-v 3464  df-in 3954

This theorem is referenced by:  inn0f  4371  csbin  4444  iunxdif3  5103  disjxun  5151  nfres  5991  nfpred  6317  cp  9934  tskwe  9993  iunconn  23423  ptclsg  23610  restmetu  24570  limciun  25914  disjunsn  32514  ordtconnlem1  33739  esum2d  33926  finminlem  36030  bj-rcleqf  36732  mbfposadd  37368  iunconnlem2  44611  disjrnmpt2  44795  disjinfi  44799  fsumiunss  45196  stoweidlem57  45678  fourierdlem80  45807  sge0iunmptlemre  46036  iundjiun  46081  pimiooltgt  46331  smflim  46398  smfpimcclem  46428  smfpimcc  46429  adddmmbl  46454  adddmmbl2  46455  muldmmbl  46456  muldmmbl2  46457  smfdivdmmbl2  46462