Theorem difeq12d 3323

Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Hypotheses

Ref	Expression
difeq12d.1	|- ( ph -> A = B )
difeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
difeq12d	|- ( ph -> ( A \ C ) = ( B \ D ) )

Proof of Theorem difeq12d

Step	Hyp	Ref	Expression
1		difeq12d.1	. . 3 |- ( ph -> A = B )
2	1	difeq1d 3321	. 2 |- ( ph -> ( A \ C ) = ( B \ C ) )
3		difeq12d.2	. . 3 |- ( ph -> C = D )
4	3	difeq2d 3322	. 2 |- ( ph -> ( B \ C ) = ( B \ D ) )
5	2, 4	eqtrd 2262	1 |- ( ph -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   = wceq 1395   \ cdif 3194

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-in1 617  ax-in2 618  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-ral 2513  df-rab 2517  df-dif 3199

This theorem is referenced by:  undifexmid  4276  exmidundif  4289  exmidundifim  4290