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

Theorem ineq12i 3334
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  |-  ( A  i^i  C )  =  ( B  i^i  D
)

Proof of Theorem ineq12i
StepHypRef Expression
1 ineq1i.1 . 2  |-  A  =  B
2 ineq12i.2 . 2  |-  C  =  D
3 ineq12 3331 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3mp2an 426 1  |-  ( A  i^i  C )  =  ( B  i^i  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1353    i^i cin 3128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135
This theorem is referenced by:  undir  3385  difindir  3390  inrab  3407  inrab2  3408  inxp  4756  resindi  4917  resindir  4918  cnvin  5031  rnin  5033  inimass  5040  funtp  5264  imainlem  5292  imain  5293  offres  6129  djuinr  7055  djuin  7056  casefun  7077  exmidfodomrlemim  7193  enq0enq  7408  explecnv  11484
  Copyright terms: Public domain W3C validator