Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfiunya GIF version

Theorem nfiunya 3685
 Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1 𝑦𝐴
nfiunya.2 𝑦𝐵
Assertion
Ref Expression
nfiunya 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiunya
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 3659 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiunya.1 . . . 4 𝑦𝐴
3 nfiunya.2 . . . . 5 𝑦𝐵
43nfcri 2172 . . . 4 𝑦 𝑧𝐵
52, 4nfrexya 2363 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2182 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2175 1 𝑦 𝑥𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  ∃wrex 2307  ∪ ciun 3657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-iun 3659 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator