Theorem ineq12d 3427

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
ineq1d.1	|- ( ph -> A = B )
ineq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
ineq12d	|- ( ph -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineq12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 |- ( ph -> A = B )
2		ineq12d.2	. 2 |- ( ph -> C = D )
3		ineq12 3421	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220

This theorem is referenced by:  csbing  3432  funprg  5411  funtpg  5412  offval  6283  ofrfval  6284  undifdc  7197  djudom  7397  ballotfilemfrc  13214  isunitd  14351  dfrhm2  14399  isrhm  14403  rhmval  14418  2idlval  14776  2idlvalg  14777  trlsegvdegfi  16588