Theorem ineq12d 3409

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 3403	. 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 1397   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  csbing  3414  funprg  5380  funtpg  5381  offval  6242  ofrfval  6243  undifdc  7115  djudom  7291  isunitd  14119  dfrhm2  14167  isrhm  14171  rhmval  14186  2idlval  14515  2idlvalg  14516