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Theorem nfin 3171
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 𝑥𝐴
nfin.2 𝑥𝐵
Assertion
Ref Expression
nfin 𝑥(𝐴𝐵)

Proof of Theorem nfin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfin5 2953 . 2 (𝐴𝐵) = {𝑦𝐴𝑦𝐵}
2 nfin.2 . . . 4 𝑥𝐵
32nfcri 2188 . . 3 𝑥 𝑦𝐵
4 nfin.1 . . 3 𝑥𝐴
53, 4nfrabxy 2507 . 2 𝑥{𝑦𝐴𝑦𝐵}
61, 5nfcxfr 2191 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 1409  wnfc 2181  {crab 2327  cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-in 2952
This theorem is referenced by:  csbing  3172  nfres  4642
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