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

Step	Hyp	Ref	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 ) )
4	1, 2, 3	mp2an 426	1 |- ( A i^i C ) = ( B i^i D )

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  4761  resindi  4922  resindir  4923  cnvin  5036  rnin  5038  inimass  5045  funtp  5269  imainlem  5297  imain  5298  offres  6135  djuinr  7061  djuin  7062  casefun  7083  exmidfodomrlemim  7199  enq0enq  7429  explecnv  11512