Theorem ineq12i 3403

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

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq12i.2	. 2 |- C = D
3		ineq12 3400	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	mp2an 426	1 |- ( A i^i C ) = ( B i^i D )

Syntax hints:   = wceq 1395   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  undir  3454  difindir  3459  inrab  3476  inrab2  3477  inxp  4856  resindi  5020  resindir  5021  cnvin  5136  rnin  5138  inimass  5145  funtp  5374  imainlem  5402  imain  5403  offres  6280  djuinr  7230  djuin  7231  casefun  7252  exmidfodomrlemim  7379  enq0enq  7618  explecnv  12016