Theorem ineq12i 3372

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 3369	. 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 1373   i^i cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  undir  3423  difindir  3428  inrab  3445  inrab2  3446  inxp  4813  resindi  4975  resindir  4976  cnvin  5091  rnin  5093  inimass  5100  funtp  5328  imainlem  5356  imain  5357  offres  6222  djuinr  7167  djuin  7168  casefun  7189  exmidfodomrlemim  7311  enq0enq  7546  explecnv  11849