Theorem difeq12d 3241

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 3239	. 2 |- ( ph -> ( A \ C ) = ( B \ C ) )
3		difeq12d.2	. . 3 |- ( ph -> C = D )
4	3	difeq2d 3240	. 2 |- ( ph -> ( B \ C ) = ( B \ D ) )
5	2, 4	eqtrd 2198	1 |- ( ph -> ( A \ C ) = ( B \ D ) )

Syntax hints:   -> wi 4   = wceq 1343   \ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-dif 3118

This theorem is referenced by:  undifexmid  4172  exmidundif  4185  exmidundifim  4186