Theorem nfin 4165

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 2147, ax-11 2163, ax-12 2185. (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 3906	. . 3 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		nfin.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfin.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1855	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∩ 𝐵)
8	7	nfci 2887	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∧ wa 395   ∈ wcel 2114  Ⅎwnfc 2884   ∩ cin 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3432  df-in 3897

This theorem is referenced by:  inn0f  4312  csbin  4383  iunxdif3  5038  disjxun  5084  nfres  5941  nfpred  6265  cp  9809  tskwe  9868  iunconn  23406  ptclsg  23593  restmetu  24548  limciun  25874  disjunsn  32682  ordtconnlem1  34087  esum2d  34256  finminlem  36519  bj-rcleqf  37351  mbfposadd  38005  iunconnlem2  45382  disjrnmpt2  45639  disjinfi  45643  fsumiunss  46026  stoweidlem57  46506  fourierdlem80  46635  sge0iunmptlemre  46864  iundjiun  46909  pimiooltgt  47159  smflim  47226  smfpimcclem  47256  smfpimcc  47257  adddmmbl  47282  adddmmbl2  47283  muldmmbl  47284  muldmmbl2  47285  smfdivdmmbl2  47290