ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq12i GIF version

Theorem ineq12i 3130
Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
ineq1i.1 A = B
ineq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
ineq12i (A𝐶) = (B𝐷)

Proof of Theorem ineq12i
StepHypRef Expression
1 ineq1i.1 . 2 A = B
2 ineq12i.2 . 2 𝐶 = 𝐷
3 ineq12 3127 . 2 ((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
41, 2, 3mp2an 402 1 (A𝐶) = (B𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  cin 2910
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918
This theorem is referenced by:  undir  3181  difindir  3186  inrab  3203  inrab2  3204  inxp  4413  resindi  4570  resindir  4571  cnvin  4674  rnin  4676  inimass  4683  funtp  4895  imainlem  4923  imain  4924  offres  5704  enq0enq  6414
  Copyright terms: Public domain W3C validator