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

Theorem ineqri 3107
 Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
ineqri.1 ((x A x B) ↔ x 𝐶)
Assertion
Ref Expression
ineqri (AB) = 𝐶
Distinct variable groups:   x,A   x,B   x,𝐶

Proof of Theorem ineqri
StepHypRef Expression
1 elin 3103 . . 3 (x (AB) ↔ (x A x B))
2 ineqri.1 . . 3 ((x A x B) ↔ x 𝐶)
31, 2bitri 173 . 2 (x (AB) ↔ x 𝐶)
43eqriv 2019 1 (AB) = 𝐶
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374   ∩ cin 2893 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901 This theorem is referenced by:  inidm  3123  inass  3124  indi  3161  inab  3182  in0  3229  pwin  3993  dmres  4559
 Copyright terms: Public domain W3C validator