Theorem nfin 4192

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 3943	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
2		nfin.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2971	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4		nfin.1	. . 3 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfrabw 3385	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
6	1, 5	nfcxfr 2975	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∈ wcel 2110  Ⅎwnfc 2961  {crab 3142   ∩ cin 3934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-in 3942

This theorem is referenced by:  csbin  4390  iunxdif3  5009  disjxun  5056  nfres  5849  nfpred  6147  cp  9314  tskwe  9373  iunconn  22030  ptclsg  22217  restmetu  23174  limciun  24486  disjunsn  30338  ordtconnlem1  31162  esum2d  31347  finminlem  33661  bj-rcleqf  34332  mbfposadd  34933  iunconnlem2  41262  inn0f  41328  disjrnmpt2  41442  disjinfi  41447  fsumiunss  41849  stoweidlem57  42336  fourierdlem80  42465  sge0iunmptlemre  42691  iundjiun  42736  pimiooltgt  42983  smflim  43047  smfpimcclem  43075  smfpimcc  43076