Theorem nfin 3951

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 3711	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
2		nfin.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2884	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4		nfin.1	. . 3 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfrab 3250	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
6	1, 5	nfcxfr 2888	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∈ wcel 2127  Ⅎwnfc 2877  {crab 3042   ∩ cin 3702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-in 3710

This theorem is referenced by:  csbin  4141  iunxdif3  4746  disjxun  4790  nfres  5541  nfpred  5834  cp  8915  tskwe  8937  iunconn  21404  ptclsg  21591  restmetu  22547  limciun  23828  disjunsn  29685  ordtconnlem1  30250  esum2d  30435  finminlem  32589  mbfposadd  33739  csbingOLD  39523  iunconnlem2  39639  inn0f  39710  disjrnmpt2  39843  disjinfi  39848  fsumiunss  40279  stoweidlem57  40746  fourierdlem80  40875  sge0iunmptlemre  41104  iundjiun  41149  pimiooltgt  41396  smflim  41460  smfpimcclem  41488  smfpimcc  41489