Theorem nfin 3782

Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

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		dfin5 3548	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
2		nfin.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2745	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4		nfin.1	. . 3 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfrab 3100	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
6	1, 5	nfcxfr 2749	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∈ wcel 1977  Ⅎwnfc 2738  {crab 2900   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547

This theorem is referenced by:  csbin  3962  iunxdif3  4542  disjxun  4581  nfres  5319  nfpred  5602  cp  8637  tskwe  8659  iuncon  21041  ptclsg  21228  restmetu  22185  limciun  23464  disjunsn  28789  ordtconlem1  29298  esum2d  29482  finminlem  31482  mbfposadd  32627  csbingOLD  38076  iunconlem2  38193  inn0f  38268  disjrnmpt2  38370  disjinfi  38375  fsumiunss  38642  stoweidlem57  38950  fourierdlem80  39079  sge0iunmptlemre  39308  iundjiun  39353  pimiooltgt  39598  smflim  39663